EJERCICIOS DE INTEGRALES INDEFINIDAS · 1. ∫x2sen 3xdx = ∫ ∫ =− − − − − xcos 3xdx 3 2 3 x cos 3x dx 3 cos 3x 2x 3 x2 cos 3x 2 = (∗) (∗) = − − −∫ dx

EJERCICIOS DE INTEGRALES INDEFINIDAS

1. ∫ dxx3senx2 = ∫ ∫−−=

−−− xdx3cosx3

2

3

x3cosxdx

3

x3cosx2

3

x3cosx 22

=

( )∗

( )∗ =

−−− ∫ dx3

x3sen

3

x3xsen

3

2

3

x3cosx2

=

++−− Kx3cos9

1

3

x3xsen

3

2

3

x3cosx2

= K27

x3cos2

9

x3xsen2

3

x3cosx2

++−−

2. ∫ +dx

9x

x2

= ∫ ++=+

K9xln2

1dx

9x

x2

2

1 22

3. ∫−

dxx1

x4

=

( ) ( )( )∫ ∫ +=

−=

−Kxarcsen

2

1dx

x1

x2

2

1dx

x1

x 2

2222

4. ( )∫ ++=+

= K1elndx1e

e)x(F x

x

x

si 2ln2K2K2ln2)0(F −=⇒=+⇒=

( ) 2ln21eln)x(F x −++=⇒

5. dx7x5

x23∫ +

−=

∫ ∫∫ =+

+−=+

+−=

++− dx

7x5

5

25

29x

5

2dx

7x5

1

5

29x

5

2dx

7x5

529

5

2

= K7x5ln25

29x

5

2 +++−

6. ( )∫ dx

x

xln 4

= ( ) Kxln5

1 5 +

7. ∫− dxex

2x3 = ∫ +−−=−−−−−

−−

K2

e

2

exdxxe2

2

1

2

ex22

2

2 xx2x

x2

3

x3cosv

xdx2du

xdx3sendv

xu 2

−=

==

=

3

x3senv

dxdu

xdx3cosdv

xu

=

==

=

2

2

x

x

2

e2

1v

xdx2du

dxxedv

xu

−

−

−=

==

=

8. ∫ dxex

1 x

1

2 = Ke x

1

+−

9. ( )∫ dxxcos = [ ]∫ ∫ ∫ ++=−== Ktcos2tsent2dtsenttsent2dttcost2dtt2tcos

=

= Kxcos2xsenx2 ++

10. ( )∫ dxxlnsenx =

∫ ∫ =−=− dx)xcos(lnx2

1

2

)x(lnsenxdx

x

1)xcos(ln

2

x

2

)x(lnsenx 222

=

+− ∫ dx

x

1)x(lnsen

2

x

2

)xcos(lnx

2

1

2

)x(lnsenx 222

=

∫−− dx)x(lnxsen4

1

4

)xcos(lnx

2

)x(lnsenx 22

I4

1

4

)xcos(lnx

2

)x(lnsenxI

22

−−=

4

)xcos(lnx

2

)x(lnsenxI

4

5 22

−=

K4

)xcos(lnx

2

)x(lnsenx

5

4dx)x(lnxsen

22

+

−=∫

11. ∫ −+

dxxx

1x2

= ∫ ∫ +−+−=−

+−K1xln2xlndx

1x

2dx

x

1

)1x(x

Bx)1x(A

1x

B

x

A

xx

1x2 −

+−=−

+=−

+

Bx)1x(A1x +−=+

2B1x

1A0x

=⇒=−=⇒=

tdt2dx

tx

tx2

==

=

sentv

dxdu

tdtcosdv

tu

===

=

2

xv

dxx

1)xcos(lndu

xdxdv

)x(lnsenu

2

=

=

==

2

xv

dxx

1)x(lnsendu

xdxdv

)xcos(lnu

2

=

−=

==

12. ∫ ∫ ∫ ∫ ∫ =++=+=+=+ − K

21

x3

25

xdxx3dxx

dxx

3dx

x

xdx

x

3x2125

212322

Kx65

xx2Kx6

5

x2 25

++=++=

13. ∫ ∫ ∫−+−=+−= dxxcosesenxexcosedxxcosexcosedxsenxe xxxxxx

IsenxexcoseI xx −+−=

senxexcoseI2 xx +−=

( )∫ ++−= Ksenxexcose2

1dxsenxe xxx

14. ∫ ∫ ∫ =−=−= dxex2

3

2

exdxx3e

2

1

2

exdxex

x22x23

2x2x23x23

=

−−= ∫ dxxe

2

ex

2

3

2

ex x2x22x23

=

−+−= ∫ dx

2

e

2

xe

2

3

4

ex3

2

ex x2x2x22x23

K8

e3

4

xe3

4

ex3

2

ex x2x2x22x23

+−+−=

15. [ ]∫ ∫ ∫ +−=+++−=+

+−

=−

K1xln2

1K1xln1xln

2

1dx

1x

1

2

1dx

1x

1

2

1dx

1x

x 22

1x

)1x(B)1x(A

1x

B

1x

A

1x

x22 −

−++=+

+−

=−

)1x(B)1x(Ax −++=

2

1AA211xSi =⇒=⇒=

2

1BB211xSi =⇒−=−⇒−=

16. ( )∫ ∫∫∫∫∫ =+

++

−+=++−+−=

+−

dx1x

12dx

1x

x

x

2xlndx

1x

2xdx

x

12dx

x

1dx

1xx

2x222222

Karctgx21xln2

1

x

2xln 2 ++−−+=

xcosv

dxeu

dxsenxdv

eu

x

x

−==

==

senxv

dxedu

dxxcosdv

eu

x

x

==

==

2

ev

dxx3du

dxedv

xu

x2

2

x2

3

=

=

=

=

2

ev

xdx2du

dxedv

xu

x2

x2

2

=

==

=

2

ev

dxdu

dxedv

xu

x2

x2

=

==

=

EJERCICIOS DE INTEGRALES INDEFINIDAS

1. ∫ dxx3senx2 = ∫ ∫−−=

−−− xdx3cosx3

2

3

x3cosxdx

3

x3cosx2

3

x3cosx 22

=

( )∗

( )∗ =

−−− ∫ dx3

x3sen

3

x3xsen

3

2

3

x3cosx2

=

++−− Kx3cos9

1

3

x3xsen

3

2

3

x3cosx2

= K27

x3cos2

9

x3xsen2

3

x3cosx2

++−−

2. ∫ +dx

9x

x2

= ∫ ++=+

K9xln2

1dx

9x

x2

2

1 22

3. ∫−

dxx1

x4

=

( ) ( )( )∫ ∫ +=

−=

−Kxarcsen

2

1dx

x1

x2

2

1dx

x1

x 2

2222

4. ( )∫ ++=+

= K1elndx1e

e)x(F x

x

x

si 2ln2K2K2ln2)0(F −=⇒=+⇒=

( ) 2ln21eln)x(F x −++=⇒

5. dx7x5

x23∫ +

−=

∫ ∫∫ =+

+−=+

+−=

++− dx

7x5

5

25

29x

5

2dx

7x5

1

5

29x

5

2dx

7x5

529

5

2

= K7x5ln25

29x

5

2 +++−

6. ( )∫ dx

x

xln 4

= ( ) Kxln5

1 5 +

7. ∫− dxex

2x3 = ∫ +−−=−−−−−

−−

K2

e

2

exdxxe2

2

1

2

ex22

2

2 xx2x

x2

3

x3cosv

xdx2du

xdx3sendv

xu 2

−=

==

=

3

x3senv

dxdu

xdx3cosdv

xu

=

==

=

2

2

x

x

2

e2

1v

xdx2du

dxxedv

xu

−

−

−=

==

=

8. ∫ dxex

1 x

1

2 = Ke x

1

+−

9. ( )∫ dxxcos = [ ]∫ ∫ ∫ ++=−== Ktcos2tsent2dtsenttsent2dttcost2dtt2tcos

=

= Kxcos2xsenx2 ++

10. ( )∫ dxxlnsenx =

∫ ∫ =−=− dx)xcos(lnx2

1

2

)x(lnsenxdx

x

1)xcos(ln

2

x

2

)x(lnsenx 222

=

+− ∫ dx

x

1)x(lnsen

2

x

2

)xcos(lnx

2

1

2

)x(lnsenx 222

=

∫−− dx)x(lnxsen4

1

4

)xcos(lnx

2

)x(lnsenx 22

I4

1

4

)xcos(lnx

2

)x(lnsenxI

22

−−=

4

)xcos(lnx

2

)x(lnsenxI

4

5 22

−=

K4

)xcos(lnx

2

)x(lnsenx

5

4dx)x(lnxsen

22

+

−=∫

11. ∫ −+

dxxx

1x2

= ∫ ∫ +−+−=−

+−K1xln2xlndx

1x

2dx

x

1

)1x(x

Bx)1x(A

1x

B

x

A

xx

1x2 −

+−=−

+=−

+

Bx)1x(A1x +−=+

2B1x

1A0x

=⇒=−=⇒=

tdt2dx

tx

tx2

==

=

sentv

dxdu

tdtcosdv

tu

===

=

2

xv

dxx

1)xcos(lndu

xdxdv

)x(lnsenu

2

=

=

==

2

xv

dxx

1)x(lnsendu

xdxdv

)xcos(lnu

2

=

−=

==

12. ∫ ∫ ∫ ∫ ∫ =++=+=+=+ − K

21

x3

25

xdxx3dxx

dxx

3dx

x

xdx

x

3x2125

212322

Kx65

xx2Kx6

5

x2 25

++=++=

13. ∫ ∫ ∫−+−=+−= dxxcosesenxexcosedxxcosexcosedxsenxe xxxxxx

IsenxexcoseI xx −+−=

senxexcoseI2 xx +−=

( )∫ ++−= Ksenxexcose2

1dxsenxe xxx

14. ∫ ∫ ∫ =−=−= dxex2

3

2

exdxx3e

2

1

2

exdxex

x22x23

2x2x23x23

=

−−= ∫ dxxe

2

ex

2

3

2

ex x2x22x23

=

−+−= ∫ dx

2

e

2

xe

2

3

4

ex3

2

ex x2x2x22x23

K8

e3

4

xe3

4

ex3

2

ex x2x2x22x23

+−+−=

15. [ ]∫ ∫ ∫ +−=+++−=+

+−

=−

K1xln2

1K1xln1xln

2

1dx

1x

1

2

1dx

1x

1

2

1dx

1x

x 22

1x

)1x(B)1x(A

1x

B

1x

A

1x

x22 −

−++=+

+−

=−

)1x(B)1x(Ax −++=

2

1AA211xSi =⇒=⇒=

2

1BB211xSi =⇒−=−⇒−=

16. ( )∫ ∫∫∫∫∫ =+

++

−+=++−+−=

+−

dx1x

12dx

1x

x

x

2xlndx

1x

2xdx

x

12dx

x

1dx

1xx

2x222222

Karctgx21xln2

1

x

2xln 2 ++−−+=

xcosv

dxeu

dxsenxdv

eu

x

x

−==

==

senxv

dxedu

dxxcosdv

eu

x

x

==

==

2

ev

dxx3du

dxedv

xu

x2

2

x2

3

=

=

=

=

2

ev

xdx2du

dxedv

xu

x2

x2

2

=

==

=

2

ev

dxdu

dxedv

xu

x2

x2

=

==

=

EJERCICIOS DE INTEGRALES INDEFINIDAS

1. ∫ dxx3senx2 = ∫ ∫−−=

−−− xdx3cosx3

2

3

x3cosxdx

3

x3cosx2

3

x3cosx 22

=

( )∗

( )∗ =

−−− ∫ dx3

x3sen

3

x3xsen

3

2

3

x3cosx2

=

++−− Kx3cos9

1

3

x3xsen

3

2

3

x3cosx2

= K27

x3cos2

9

x3xsen2

3

x3cosx2

++−−

2. ∫ +dx

9x

x2

= ∫ ++=+

K9xln2

1dx

9x

x2

2

1 22

3. ∫−

dxx1

x4

=

( ) ( )( )∫ ∫ +=

−=

−Kxarcsen

2

1dx

x1

x2

2

1dx

x1

x 2

2222

4. ( )∫ ++=+

= K1elndx1e

e)x(F x

x

x

si 2ln2K2K2ln2)0(F −=⇒=+⇒=

( ) 2ln21eln)x(F x −++=⇒

5. dx7x5

x23∫ +

−=

∫ ∫∫ =+

+−=+

+−=

++− dx

7x5

5

25

29x

5

2dx

7x5

1

5

29x

5

2dx

7x5

529

5

2

= K7x5ln25

29x

5

2 +++−

6. ( )∫ dx

x

xln 4

= ( ) Kxln5

1 5 +

7. ∫− dxex

2x3 = ∫ +−−=−−−−−

−−

K2

e

2

exdxxe2

2

1

2

ex22

2

2 xx2x

x2

3

x3cosv

xdx2du

xdx3sendv

xu 2

−=

==

=

3

x3senv

dxdu

xdx3cosdv

xu

=

==

=

2

2

x

x

2

e2

1v

xdx2du

dxxedv

xu

−

−

−=

==

=

8. ∫ dxex

1 x

1

2 = Ke x

1

+−

9. ( )∫ dxxcos = [ ]∫ ∫ ∫ ++=−== Ktcos2tsent2dtsenttsent2dttcost2dtt2tcos

=

= Kxcos2xsenx2 ++

10. ( )∫ dxxlnsenx =

∫ ∫ =−=− dx)xcos(lnx2

1

2

)x(lnsenxdx

x

1)xcos(ln

2

x

2

)x(lnsenx 222

=

+− ∫ dx

x

1)x(lnsen

2

x

2

)xcos(lnx

2

1

2

)x(lnsenx 222

=

∫−− dx)x(lnxsen4

1

4

)xcos(lnx

2

)x(lnsenx 22

I4

1

4

)xcos(lnx

2

)x(lnsenxI

22

−−=

4

)xcos(lnx

2

)x(lnsenxI

4

5 22

−=

K4

)xcos(lnx

2

)x(lnsenx

5

4dx)x(lnxsen

22

+

−=∫

11. ∫ −+

dxxx

1x2

= ∫ ∫ +−+−=−

+−K1xln2xlndx

1x

2dx

x

1

)1x(x

Bx)1x(A

1x

B

x

A

xx

1x2 −

+−=−

+=−

+

Bx)1x(A1x +−=+

2B1x

1A0x

=⇒=−=⇒=

tdt2dx

tx

tx2

==

=

sentv

dxdu

tdtcosdv

tu

===

=

2

xv

dxx

1)xcos(lndu

xdxdv

)x(lnsenu

2

=

=

==

2

xv

dxx

1)x(lnsendu

xdxdv

)xcos(lnu

2

=

−=

==

12. ∫ ∫ ∫ ∫ ∫ =++=+=+=+ − K

21

x3

25

xdxx3dxx

dxx

3dx

x

xdx

x

3x2125

212322

Kx65

xx2Kx6

5

x2 25

++=++=

13. ∫ ∫ ∫−+−=+−= dxxcosesenxexcosedxxcosexcosedxsenxe xxxxxx

IsenxexcoseI xx −+−=

senxexcoseI2 xx +−=

( )∫ ++−= Ksenxexcose2

1dxsenxe xxx

14. ∫ ∫ ∫ =−=−= dxex2

3

2

exdxx3e

2

1

2

exdxex

x22x23

2x2x23x23

=

−−= ∫ dxxe

2

ex

2

3

2

ex x2x22x23

=

−+−= ∫ dx

2

e

2

xe

2

3

4

ex3

2

ex x2x2x22x23

K8

e3

4

xe3

4

ex3

2

ex x2x2x22x23

+−+−=

15. [ ]∫ ∫ ∫ +−=+++−=+

+−

=−

K1xln2

1K1xln1xln

2

1dx

1x

1

2

1dx

1x

1

2

1dx

1x

x 22

1x

)1x(B)1x(A

1x

B

1x

A

1x

x22 −

−++=+

+−

=−

)1x(B)1x(Ax −++=

2

1AA211xSi =⇒=⇒=

2

1BB211xSi =⇒−=−⇒−=

16. ( )∫ ∫∫∫∫∫ =+

++

−+=++−+−=

+−

dx1x

12dx

1x

x

x

2xlndx

1x

2xdx

x

12dx

x

1dx

1xx

2x222222

Karctgx21xln2

1

x

2xln 2 ++−−+=

xcosv

dxeu

dxsenxdv

eu

x

x

−==

==

senxv

dxedu

dxxcosdv

eu

x

x

==

==

2

ev

dxx3du

dxedv

xu

x2

2

x2

3

=

=

=

=

2

ev

xdx2du

dxedv

xu

x2

x2

2

=

==

=

2

ev

dxdu

dxedv

xu

x2

x2

=

==

=

EJERCICIOS DE INTEGRALES INDEFINIDAS

1. ∫ dxx3senx2 = ∫ ∫−−=

−−− xdx3cosx3

2

3

x3cosxdx

3

x3cosx2

3

x3cosx 22

=

( )∗

( )∗ =

−−− ∫ dx3

x3sen

3

x3xsen

3

2

3

x3cosx2

=

++−− Kx3cos9

1

3

x3xsen

3

2

3

x3cosx2

= K27

x3cos2

9

x3xsen2

3

x3cosx2

++−−

2. ∫ +dx

9x

x2

= ∫ ++=+

K9xln2

1dx

9x

x2

2

1 22

3. ∫−

dxx1

x4

=

( ) ( )( )∫ ∫ +=

−=

−Kxarcsen

2

1dx

x1

x2

2

1dx

x1

x 2

2222

4. ( )∫ ++=+

= K1elndx1e

e)x(F x

x

x

si 2ln2K2K2ln2)0(F −=⇒=+⇒=

( ) 2ln21eln)x(F x −++=⇒

5. dx7x5

x23∫ +

−=

∫ ∫∫ =+

+−=+

+−=

++− dx

7x5

5

25

29x

5

2dx

7x5

1

5

29x

5

2dx

7x5

529

5

2

= K7x5ln25

29x

5

2 +++−

6. ( )∫ dx

x

xln 4

= ( ) Kxln5

1 5 +

7. ∫− dxex

2x3 = ∫ +−−=−−−−−

−−

K2

e

2

exdxxe2

2

1

2

ex22

2

2 xx2x

x2

3

x3cosv

xdx2du

xdx3sendv

xu 2

−=

==

=

3

x3senv

dxdu

xdx3cosdv

xu

=

==

=

2

2

x

x

2

e2

1v

xdx2du

dxxedv

xu

−

−

−=

==

=

8. ∫ dxex

1 x

1

2 = Ke x

1

+−

9. ( )∫ dxxcos = [ ]∫ ∫ ∫ ++=−== Ktcos2tsent2dtsenttsent2dttcost2dtt2tcos

=

= Kxcos2xsenx2 ++

10. ( )∫ dxxlnsenx =

∫ ∫ =−=− dx)xcos(lnx2

1

2

)x(lnsenxdx

x

1)xcos(ln

2

x

2

)x(lnsenx 222

=

+− ∫ dx

x

1)x(lnsen

2

x

2

)xcos(lnx

2

1

2

)x(lnsenx 222

=

∫−− dx)x(lnxsen4

1

4

)xcos(lnx

2

)x(lnsenx 22

I4

1

4

)xcos(lnx

2

)x(lnsenxI

22

−−=

4

)xcos(lnx

2

)x(lnsenxI

4

5 22

−=

K4

)xcos(lnx

2

)x(lnsenx

5

4dx)x(lnxsen

22

+

−=∫

11. ∫ −+

dxxx

1x2

= ∫ ∫ +−+−=−

+−K1xln2xlndx

1x

2dx

x

1

)1x(x

Bx)1x(A

1x

B

x

A

xx

1x2 −

+−=−

+=−

+

Bx)1x(A1x +−=+

2B1x

1A0x

=⇒=−=⇒=

tdt2dx

tx

tx2

==

=

sentv

dxdu

tdtcosdv

tu

===

=

2

xv

dxx

1)xcos(lndu

xdxdv

)x(lnsenu

2

=

=

==

2

xv

dxx

1)x(lnsendu

xdxdv

)xcos(lnu

2

=

−=

==

12. ∫ ∫ ∫ ∫ ∫ =++=+=+=+ − K

21

x3

25

xdxx3dxx

dxx

3dx

x

xdx

x

3x2125

212322

Kx65

xx2Kx6

5

x2 25

++=++=

13. ∫ ∫ ∫−+−=+−= dxxcosesenxexcosedxxcosexcosedxsenxe xxxxxx

IsenxexcoseI xx −+−=

senxexcoseI2 xx +−=

( )∫ ++−= Ksenxexcose2

1dxsenxe xxx

14. ∫ ∫ ∫ =−=−= dxex2

3

2

exdxx3e

2

1

2

exdxex

x22x23

2x2x23x23

=

−−= ∫ dxxe

2

ex

2

3

2

ex x2x22x23

=

−+−= ∫ dx

2

e

2

xe

2

3

4

ex3

2

ex x2x2x22x23

K8

e3

4

xe3

4

ex3

2

ex x2x2x22x23

+−+−=

15. [ ]∫ ∫ ∫ +−=+++−=+

+−

=−

K1xln2

1K1xln1xln

2

1dx

1x

1

2

1dx

1x

1

2

1dx

1x

x 22

1x

)1x(B)1x(A

1x

B

1x

A

1x

x22 −

−++=+

+−

=−

)1x(B)1x(Ax −++=

2

1AA211xSi =⇒=⇒=

2

1BB211xSi =⇒−=−⇒−=

16. ( )∫ ∫∫∫∫∫ =+

++

−+=++−+−=

+−

dx1x

12dx

1x

x

x

2xlndx

1x

2xdx

x

12dx

x

1dx

1xx

2x222222

Karctgx21xln2

1

x

2xln 2 ++−−+=

xcosv

dxeu

dxsenxdv

eu

x

x

−==

==

senxv

dxedu

dxxcosdv

eu

x

x

==

==

2

ev

dxx3du

dxedv

xu

x2

2

x2

3

=

=

=

=

2

ev

xdx2du

dxedv

xu

x2

x2

2

=

==

=

2

ev

dxdu

dxedv

xu

x2

x2

=

==

=

EJERCICIOS DE INTEGRALES INDEFINIDAS

1. ∫ dxx3senx2 = ∫ ∫−−=

−−− xdx3cosx3

2

3

x3cosxdx

3

x3cosx2

3

x3cosx 22

=

( )∗

( )∗ =

−−− ∫ dx3

x3sen

3

x3xsen

3

2

3

x3cosx2

=

++−− Kx3cos9

1

3

x3xsen

3

2

3

x3cosx2

= K27

x3cos2

9

x3xsen2

3

x3cosx2

++−−

2. ∫ +dx

9x

x2

= ∫ ++=+

K9xln2

1dx

9x

x2

2

1 22

3. ∫−

dxx1

x4

=

( ) ( )( )∫ ∫ +=

−=

−Kxarcsen

2

1dx

x1

x2

2

1dx

x1

x 2

2222

4. ( )∫ ++=+

= K1elndx1e

e)x(F x

x

x

si 2ln2K2K2ln2)0(F −=⇒=+⇒=

( ) 2ln21eln)x(F x −++=⇒

5. dx7x5

x23∫ +

−=

∫ ∫∫ =+

+−=+

+−=

++− dx

7x5

5

25

29x

5

2dx

7x5

1

5

29x

5

2dx

7x5

529

5

2

= K7x5ln25

29x

5

2 +++−

6. ( )∫ dx

x

xln 4

= ( ) Kxln5

1 5 +

7. ∫− dxex

2x3 = ∫ +−−=−−−−−

−−

K2

e

2

exdxxe2

2

1

2

ex22

2

2 xx2x

x2

3

x3cosv

xdx2du

xdx3sendv

xu 2

−=

==

=

3

x3senv

dxdu

xdx3cosdv

xu

=

==

=

2

2

x

x

2

e2

1v

xdx2du

dxxedv

xu

−

−

−=

==

=

8. ∫ dxex

1 x

1

2 = Ke x

1

+−

9. ( )∫ dxxcos = [ ]∫ ∫ ∫ ++=−== Ktcos2tsent2dtsenttsent2dttcost2dtt2tcos

=

= Kxcos2xsenx2 ++

10. ( )∫ dxxlnsenx =

∫ ∫ =−=− dx)xcos(lnx2

1

2

)x(lnsenxdx

x

1)xcos(ln

2

x

2

)x(lnsenx 222

=

+− ∫ dx

x

1)x(lnsen

2

x

2

)xcos(lnx

2

1

2

)x(lnsenx 222

=

∫−− dx)x(lnxsen4

1

4

)xcos(lnx

2

)x(lnsenx 22

I4

1

4

)xcos(lnx

2

)x(lnsenxI

22

−−=

4

)xcos(lnx

2

)x(lnsenxI

4

5 22

−=

K4

)xcos(lnx

2

)x(lnsenx

5

4dx)x(lnxsen

22

+

−=∫

11. ∫ −+

dxxx

1x2

= ∫ ∫ +−+−=−

+−K1xln2xlndx

1x

2dx

x

1

)1x(x

Bx)1x(A

1x

B

x

A

xx

1x2 −

+−=−

+=−

+

Bx)1x(A1x +−=+

2B1x

1A0x

=⇒=−=⇒=

tdt2dx

tx

tx2

==

=

sentv

dxdu

tdtcosdv

tu

===

=

2

xv

dxx

1)xcos(lndu

xdxdv

)x(lnsenu

2

=

=

==

2

xv

dxx

1)x(lnsendu

xdxdv

)xcos(lnu

2

=

−=

==

12. ∫ ∫ ∫ ∫ ∫ =++=+=+=+ − K

21

x3

25

xdxx3dxx

dxx

3dx

x

xdx

x

3x2125

212322

Kx65

xx2Kx6

5

x2 25

++=++=

13. ∫ ∫ ∫−+−=+−= dxxcosesenxexcosedxxcosexcosedxsenxe xxxxxx

IsenxexcoseI xx −+−=

senxexcoseI2 xx +−=

( )∫ ++−= Ksenxexcose2

1dxsenxe xxx

14. ∫ ∫ ∫ =−=−= dxex2

3

2

exdxx3e

2

1

2

exdxex

x22x23

2x2x23x23

=

−−= ∫ dxxe

2

ex

2

3

2

ex x2x22x23

=

−+−= ∫ dx

2

e

2

xe

2

3

4

ex3

2

ex x2x2x22x23

K8

e3

4

xe3

4

ex3

2

ex x2x2x22x23

+−+−=

15. [ ]∫ ∫ ∫ +−=+++−=+

+−

=−

K1xln2

1K1xln1xln

2

1dx

1x

1

2

1dx

1x

1

2

1dx

1x

x 22

1x

)1x(B)1x(A

1x

B

1x

A

1x

x22 −

−++=+

+−

=−

)1x(B)1x(Ax −++=

2

1AA211xSi =⇒=⇒=

2

1BB211xSi =⇒−=−⇒−=

16. ( )∫ ∫∫∫∫∫ =+

++

−+=++−+−=

+−

dx1x

12dx

1x

x

x

2xlndx

1x

2xdx

x

12dx

x

1dx

1xx

2x222222

Karctgx21xln2

1

x

2xln 2 ++−−+=

xcosv

dxeu

dxsenxdv

eu

x

x

−==

==

senxv

dxedu

dxxcosdv

eu

x

x

==

==

2

ev

dxx3du

dxedv

xu

x2

2

x2

3

=

=

=

=

2

ev

xdx2du

dxedv

xu

x2

x2

2

=

==

=

2

ev

dxdu

dxedv

xu

x2

x2

=

==

=

EJERCICIOS DE INTEGRALES INDEFINIDAS

1. ∫ dxx3senx2 = ∫ ∫−−=

−−− xdx3cosx3

2

3

x3cosxdx

3

x3cosx2

3

x3cosx 22

=

( )∗

( )∗ =

−−− ∫ dx3

x3sen

3

x3xsen

3

2

3

x3cosx2

=

++−− Kx3cos9

1

3

x3xsen

3

2

3

x3cosx2

= K27

x3cos2

9

x3xsen2

3

x3cosx2

++−−

2. ∫ +dx

9x

x2

= ∫ ++=+

K9xln2

1dx

9x

x2

2

1 22

3. ∫−

dxx1

x4

=

( ) ( )( )∫ ∫ +=

−=

−Kxarcsen

2

1dx

x1

x2

2

1dx

x1

x 2

2222

4. ( )∫ ++=+

= K1elndx1e

e)x(F x

x

x

si 2ln2K2K2ln2)0(F −=⇒=+⇒=

( ) 2ln21eln)x(F x −++=⇒

5. dx7x5

x23∫ +

−=

∫ ∫∫ =+

+−=+

+−=

++− dx

7x5

5

25

29x

5

2dx

7x5

1

5

29x

5

2dx

7x5

529

5

2

= K7x5ln25

29x

5

2 +++−

6. ( )∫ dx

x

xln 4

= ( ) Kxln5

1 5 +

7. ∫− dxex

2x3 = ∫ +−−=−−−−−

−−

K2

e

2

exdxxe2

2

1

2

ex22

2

2 xx2x

x2

3

x3cosv

xdx2du

xdx3sendv

xu 2

−=

==

=

3

x3senv

dxdu

xdx3cosdv

xu

=

==

=

2

2

x

x

2

e2

1v

xdx2du

dxxedv

xu

−

−

−=

==

=

8. ∫ dxex

1 x

1

2 = Ke x

1

+−

9. ( )∫ dxxcos = [ ]∫ ∫ ∫ ++=−== Ktcos2tsent2dtsenttsent2dttcost2dtt2tcos

=

= Kxcos2xsenx2 ++

10. ( )∫ dxxlnsenx =

∫ ∫ =−=− dx)xcos(lnx2

1

2

)x(lnsenxdx

x

1)xcos(ln

2

x

2

)x(lnsenx 222

=

+− ∫ dx

x

1)x(lnsen

2

x

2

)xcos(lnx

2

1

2

)x(lnsenx 222

=

∫−− dx)x(lnxsen4

1

4

)xcos(lnx

2

)x(lnsenx 22

I4

1

4

)xcos(lnx

2

)x(lnsenxI

22

−−=

4

)xcos(lnx

2

)x(lnsenxI

4

5 22

−=

K4

)xcos(lnx

2

)x(lnsenx

5

4dx)x(lnxsen

22

+

−=∫

11. ∫ −+

dxxx

1x2

= ∫ ∫ +−+−=−

+−K1xln2xlndx

1x

2dx

x

1

)1x(x

Bx)1x(A

1x

B

x

A

xx

1x2 −

+−=−

+=−

+

Bx)1x(A1x +−=+

2B1x

1A0x

=⇒=−=⇒=

tdt2dx

tx

tx2

==

=

sentv

dxdu

tdtcosdv

tu

===

=

2

xv

dxx

1)xcos(lndu

xdxdv

)x(lnsenu

2

=

=

==

2

xv

dxx

1)x(lnsendu

xdxdv

)xcos(lnu

2

=

−=

==

12. ∫ ∫ ∫ ∫ ∫ =++=+=+=+ − K

21

x3

25

xdxx3dxx

dxx

3dx

x

xdx

x

3x2125

212322

Kx65

xx2Kx6

5

x2 25

++=++=

13. ∫ ∫ ∫−+−=+−= dxxcosesenxexcosedxxcosexcosedxsenxe xxxxxx

IsenxexcoseI xx −+−=

senxexcoseI2 xx +−=

( )∫ ++−= Ksenxexcose2

1dxsenxe xxx

14. ∫ ∫ ∫ =−=−= dxex2

3

2

exdxx3e

2

1

2

exdxex

x22x23

2x2x23x23

=

−−= ∫ dxxe

2

ex

2

3

2

ex x2x22x23

=

−+−= ∫ dx

2

e

2

xe

2

3

4

ex3

2

ex x2x2x22x23

K8

e3

4

xe3

4

ex3

2

ex x2x2x22x23

+−+−=

15. [ ]∫ ∫ ∫ +−=+++−=+

+−

=−

K1xln2

1K1xln1xln

2

1dx

1x

1

2

1dx

1x

1

2

1dx

1x

x 22

1x

)1x(B)1x(A

1x

B

1x

A

1x

x22 −

−++=+

+−

=−

)1x(B)1x(Ax −++=

2

1AA211xSi =⇒=⇒=

2

1BB211xSi =⇒−=−⇒−=

16. ( )∫ ∫∫∫∫∫ =+

++

−+=++−+−=

+−

dx1x

12dx

1x

x

x

2xlndx

1x

2xdx

x

12dx

x

1dx

1xx

2x222222

Karctgx21xln2

1

x

2xln 2 ++−−+=

xcosv

dxeu

dxsenxdv

eu

x

x

−==

==

senxv

dxedu

dxxcosdv

eu

x

x

==

==

2

ev

dxx3du

dxedv

xu

x2

2

x2

3

=

=

=

=

2

ev

xdx2du

dxedv

xu

x2

x2

2

=

==

=

2

ev

dxdu

dxedv

xu

x2

x2

=

==

=

EJERCICIOS DE INTEGRALES INDEFINIDAS

1. ∫ dxx3senx2 = ∫ ∫−−=

−−− xdx3cosx3

2

3

x3cosxdx

3

x3cosx2

3

x3cosx 22

=

( )∗

( )∗ =

−−− ∫ dx3

x3sen

3

x3xsen

3

2

3

x3cosx2

=

++−− Kx3cos9

1

3

x3xsen

3

2

3

x3cosx2

= K27

x3cos2

9

x3xsen2

3

x3cosx2

++−−

2. ∫ +dx

9x

x2

= ∫ ++=+

K9xln2

1dx

9x

x2

2

1 22

3. ∫−

dxx1

x4

=

( ) ( )( )∫ ∫ +=

−=

−Kxarcsen

2

1dx

x1

x2

2

1dx

x1

x 2

2222

4. ( )∫ ++=+

= K1elndx1e

e)x(F x

x

x

si 2ln2K2K2ln2)0(F −=⇒=+⇒=

( ) 2ln21eln)x(F x −++=⇒

5. dx7x5

x23∫ +

−=

∫ ∫∫ =+

+−=+

+−=

++− dx

7x5

5

25

29x

5

2dx

7x5

1

5

29x

5

2dx

7x5

529

5

2

= K7x5ln25

29x

5

2 +++−

6. ( )∫ dx

x

xln 4

= ( ) Kxln5

1 5 +

7. ∫− dxex

2x3 = ∫ +−−=−−−−−

−−

K2

e

2

exdxxe2

2

1

2

ex22

2

2 xx2x

x2

3

x3cosv

xdx2du

xdx3sendv

xu 2

−=

==

=

3

x3senv

dxdu

xdx3cosdv

xu

=

==

=

2

2

x

x

2

e2

1v

xdx2du

dxxedv

xu

−

−

−=

==

=

8. ∫ dxex

1 x

1

2 = Ke x

1

+−

9. ( )∫ dxxcos = [ ]∫ ∫ ∫ ++=−== Ktcos2tsent2dtsenttsent2dttcost2dtt2tcos

=

= Kxcos2xsenx2 ++

10. ( )∫ dxxlnsenx =

∫ ∫ =−=− dx)xcos(lnx2

1

2

)x(lnsenxdx

x

1)xcos(ln

2

x

2

)x(lnsenx 222

=

+− ∫ dx

x

1)x(lnsen

2

x

2

)xcos(lnx

2

1

2

)x(lnsenx 222

=

∫−− dx)x(lnxsen4

1

4

)xcos(lnx

2

)x(lnsenx 22

I4

1

4

)xcos(lnx

2

)x(lnsenxI

22

−−=

4

)xcos(lnx

2

)x(lnsenxI

4

5 22

−=

K4

)xcos(lnx

2

)x(lnsenx

5

4dx)x(lnxsen

22

+

−=∫

11. ∫ −+

dxxx

1x2

= ∫ ∫ +−+−=−

+−K1xln2xlndx

1x

2dx

x

1

)1x(x

Bx)1x(A

1x

B

x

A

xx

1x2 −

+−=−

+=−

+

Bx)1x(A1x +−=+

2B1x

1A0x

=⇒=−=⇒=

tdt2dx

tx

tx2

==

=

sentv

dxdu

tdtcosdv

tu

===

=

2

xv

dxx

1)xcos(lndu

xdxdv

)x(lnsenu

2

=

=

==

2

xv

dxx

1)x(lnsendu

xdxdv

)xcos(lnu

2

=

−=

==

12. ∫ ∫ ∫ ∫ ∫ =++=+=+=+ − K

21

x3

25

xdxx3dxx

dxx

3dx

x

xdx

x

3x2125

212322

Kx65

xx2Kx6

5

x2 25

++=++=

13. ∫ ∫ ∫−+−=+−= dxxcosesenxexcosedxxcosexcosedxsenxe xxxxxx

IsenxexcoseI xx −+−=

senxexcoseI2 xx +−=

( )∫ ++−= Ksenxexcose2

1dxsenxe xxx

14. ∫ ∫ ∫ =−=−= dxex2

3

2

exdxx3e

2

1

2

exdxex

x22x23

2x2x23x23

=

−−= ∫ dxxe

2

ex

2

3

2

ex x2x22x23

=

−+−= ∫ dx

2

e

2

xe

2

3

4

ex3

2

ex x2x2x22x23

K8

e3

4

xe3

4

ex3

2

ex x2x2x22x23

+−+−=

15. [ ]∫ ∫ ∫ +−=+++−=+

+−

=−

K1xln2

1K1xln1xln

2

1dx

1x

1

2

1dx

1x

1

2

1dx

1x

x 22

1x

)1x(B)1x(A

1x

B

1x

A

1x

x22 −

−++=+

+−

=−

)1x(B)1x(Ax −++=

2

1AA211xSi =⇒=⇒=

2

1BB211xSi =⇒−=−⇒−=

16. ( )∫ ∫∫∫∫∫ =+

++

−+=++−+−=

+−

dx1x

12dx

1x

x

x

2xlndx

1x

2xdx

x

12dx

x

1dx

1xx

2x222222

Karctgx21xln2

1

x

2xln 2 ++−−+=

xcosv

dxeu

dxsenxdv

eu

x

x

−==

==

senxv

dxedu

dxxcosdv

eu

x

x

==

==

2

ev

dxx3du

dxedv

xu

x2

2

x2

3

=

=

=

=

2

ev

xdx2du

dxedv

xu

x2

x2

2

=

==

=

2

ev

dxdu

dxedv

xu

x2

x2

=

==

=

EJERCICIOS DE INTEGRALES INDEFINIDAS

1. ∫ dxx3senx2 = ∫ ∫−−=

−−− xdx3cosx3

2

3

x3cosxdx

3

x3cosx2

3

x3cosx 22

=

( )∗

( )∗ =

−−− ∫ dx3

x3sen

3

x3xsen

3

2

3

x3cosx2

=

++−− Kx3cos9

1

3

x3xsen

3

2

3

x3cosx2

= K27

x3cos2

9

x3xsen2

3

x3cosx2

++−−

2. ∫ +dx

9x

x2

= ∫ ++=+

K9xln2

1dx

9x

x2

2

1 22

3. ∫−

dxx1

x4

=

( ) ( )( )∫ ∫ +=

−=

−Kxarcsen

2

1dx

x1

x2

2

1dx

x1

x 2

2222

4. ( )∫ ++=+

= K1elndx1e

e)x(F x

x

x

si 2ln2K2K2ln2)0(F −=⇒=+⇒=

( ) 2ln21eln)x(F x −++=⇒

5. dx7x5

x23∫ +

−=

∫ ∫∫ =+

+−=+

+−=

++− dx

7x5

5

25

29x

5

2dx

7x5

1

5

29x

5

2dx

7x5

529

5

2

= K7x5ln25

29x

5

2 +++−

6. ( )∫ dx

x

xln 4

= ( ) Kxln5

1 5 +

7. ∫− dxex

2x3 = ∫ +−−=−−−−−

−−

K2

e

2

exdxxe2

2

1

2

ex22

2

2 xx2x

x2

3

x3cosv

xdx2du

xdx3sendv

xu 2

−=

==

=

3

x3senv

dxdu

xdx3cosdv

xu

=

==

=

2

2

x

x

2

e2

1v

xdx2du

dxxedv

xu

−

−

−=

==

=

8. ∫ dxex

1 x

1

2 = Ke x

1

+−

9. ( )∫ dxxcos = [ ]∫ ∫ ∫ ++=−== Ktcos2tsent2dtsenttsent2dttcost2dtt2tcos

=

= Kxcos2xsenx2 ++

10. ( )∫ dxxlnsenx =

∫ ∫ =−=− dx)xcos(lnx2

1

2

)x(lnsenxdx

x

1)xcos(ln

2

x

2

)x(lnsenx 222

=

+− ∫ dx

x

1)x(lnsen

2

x

2

)xcos(lnx

2

1

2

)x(lnsenx 222

=

∫−− dx)x(lnxsen4

1

4

)xcos(lnx

2

)x(lnsenx 22

I4

1

4

)xcos(lnx

2

)x(lnsenxI

22

−−=

4

)xcos(lnx

2

)x(lnsenxI

4

5 22

−=

K4

)xcos(lnx

2

)x(lnsenx

5

4dx)x(lnxsen

22

+

−=∫

11. ∫ −+

dxxx

1x2

= ∫ ∫ +−+−=−

+−K1xln2xlndx

1x

2dx

x

1

)1x(x

Bx)1x(A

1x

B

x

A

xx

1x2 −

+−=−

+=−

+

Bx)1x(A1x +−=+

2B1x

1A0x

=⇒=−=⇒=

tdt2dx

tx

tx2

==

=

sentv

dxdu

tdtcosdv

tu

===

=

2

xv

dxx

1)xcos(lndu

xdxdv

)x(lnsenu

2

=

=

==

2

xv

dxx

1)x(lnsendu

xdxdv

)xcos(lnu

2

=

−=

==

12. ∫ ∫ ∫ ∫ ∫ =++=+=+=+ − K

21

x3

25

xdxx3dxx

dxx

3dx

x

xdx

x

3x2125

212322

Kx65

xx2Kx6

5

x2 25

++=++=

13. ∫ ∫ ∫−+−=+−= dxxcosesenxexcosedxxcosexcosedxsenxe xxxxxx

IsenxexcoseI xx −+−=

senxexcoseI2 xx +−=

( )∫ ++−= Ksenxexcose2

1dxsenxe xxx

14. ∫ ∫ ∫ =−=−= dxex2

3

2

exdxx3e

2

1

2

exdxex

x22x23

2x2x23x23

=

−−= ∫ dxxe

2

ex

2

3

2

ex x2x22x23

=

−+−= ∫ dx

2

e

2

xe

2

3

4

ex3

2

ex x2x2x22x23

K8

e3

4

xe3

4

ex3

2

ex x2x2x22x23

+−+−=

15. [ ]∫ ∫ ∫ +−=+++−=+

+−

=−

K1xln2

1K1xln1xln

2

1dx

1x

1

2

1dx

1x

1

2

1dx

1x

x 22

1x

)1x(B)1x(A

1x

B

1x

A

1x

x22 −

−++=+

+−

=−

)1x(B)1x(Ax −++=

2

1AA211xSi =⇒=⇒=

2

1BB211xSi =⇒−=−⇒−=

16. ( )∫ ∫∫∫∫∫ =+

++

−+=++−+−=

+−

dx1x

12dx

1x

x

x

2xlndx

1x

2xdx

x

12dx

x

1dx

1xx

2x222222

Karctgx21xln2

1

x

2xln 2 ++−−+=

xcosv

dxeu

dxsenxdv

eu

x

x

−==

==

senxv

dxedu

dxxcosdv

eu

x

x

==

==

2

ev

dxx3du

dxedv

xu

x2

2

x2

3

=

=

=

=

2

ev

xdx2du

dxedv

xu

x2

x2

2

=

==

=

2

ev

dxdu

dxedv

xu

x2

x2

=

==

=

EJERCICIOS DE INTEGRALES INDEFINIDAS

1. ∫ dxx3senx2 = ∫ ∫−−=

−−− xdx3cosx3

2

3

x3cosxdx

3

x3cosx2

3

x3cosx 22

=

( )∗

( )∗ =

−−− ∫ dx3

x3sen

3

x3xsen

3

2

3

x3cosx2

=

++−− Kx3cos9

1

3

x3xsen

3

2

3

x3cosx2

= K27

x3cos2

9

x3xsen2

3

x3cosx2

++−−

2. ∫ +dx

9x

x2

= ∫ ++=+

K9xln2

1dx

9x

x2

2

1 22

3. ∫−

dxx1

x4

=

( ) ( )( )∫ ∫ +=

−=

−Kxarcsen

2

1dx

x1

x2

2

1dx

x1

x 2

2222

4. ( )∫ ++=+

= K1elndx1e

e)x(F x

x

x

si 2ln2K2K2ln2)0(F −=⇒=+⇒=

( ) 2ln21eln)x(F x −++=⇒

5. dx7x5

x23∫ +

−=

∫ ∫∫ =+

+−=+

+−=

++− dx

7x5

5

25

29x

5

2dx

7x5

1

5

29x

5

2dx

7x5

529

5

2

= K7x5ln25

29x

5

2 +++−

6. ( )∫ dx

x

xln 4

= ( ) Kxln5

1 5 +

7. ∫− dxex

2x3 = ∫ +−−=−−−−−

−−

K2

e

2

exdxxe2

2

1

2

ex22

2

2 xx2x

x2

3

x3cosv

xdx2du

xdx3sendv

xu 2

−=

==

=

3

x3senv

dxdu

xdx3cosdv

xu

=

==

=

2

2

x

x

2

e2

1v

xdx2du

dxxedv

xu

−

−

−=

==

=

8. ∫ dxex

1 x

1

2 = Ke x

1

+−

9. ( )∫ dxxcos = [ ]∫ ∫ ∫ ++=−== Ktcos2tsent2dtsenttsent2dttcost2dtt2tcos

=

= Kxcos2xsenx2 ++

10. ( )∫ dxxlnsenx =

∫ ∫ =−=− dx)xcos(lnx2

1

2

)x(lnsenxdx

x

1)xcos(ln

2

x

2

)x(lnsenx 222

=

+− ∫ dx

x

1)x(lnsen

2

x

2

)xcos(lnx

2

1

2

)x(lnsenx 222

=

∫−− dx)x(lnxsen4

1

4

)xcos(lnx

2

)x(lnsenx 22

I4

1

4

)xcos(lnx

2

)x(lnsenxI

22

−−=

4

)xcos(lnx

2

)x(lnsenxI

4

5 22

−=

K4

)xcos(lnx

2

)x(lnsenx

5

4dx)x(lnxsen

22

+

−=∫

11. ∫ −+

dxxx

1x2

= ∫ ∫ +−+−=−

+−K1xln2xlndx

1x

2dx

x

1

)1x(x

Bx)1x(A

1x

B

x

A

xx

1x2 −

+−=−

+=−

+

Bx)1x(A1x +−=+

2B1x

1A0x

=⇒=−=⇒=

tdt2dx

tx

tx2

==

=

sentv

dxdu

tdtcosdv

tu

===

=

2

xv

dxx

1)xcos(lndu

xdxdv

)x(lnsenu

2

=

=

==

2

xv

dxx

1)x(lnsendu

xdxdv

)xcos(lnu

2

=

−=

==

12. ∫ ∫ ∫ ∫ ∫ =++=+=+=+ − K

21

x3

25

xdxx3dxx

dxx

3dx

x

xdx

x

3x2125

212322

Kx65

xx2Kx6

5

x2 25

++=++=

13. ∫ ∫ ∫−+−=+−= dxxcosesenxexcosedxxcosexcosedxsenxe xxxxxx

IsenxexcoseI xx −+−=

senxexcoseI2 xx +−=

( )∫ ++−= Ksenxexcose2

1dxsenxe xxx

14. ∫ ∫ ∫ =−=−= dxex2

3

2

exdxx3e

2

1

2

exdxex

x22x23

2x2x23x23

=

−−= ∫ dxxe

2

ex

2

3

2

ex x2x22x23

=

−+−= ∫ dx

2

e

2

xe

2

3

4

ex3

2

ex x2x2x22x23

K8

e3

4

xe3

4

ex3

2

ex x2x2x22x23

+−+−=

15. [ ]∫ ∫ ∫ +−=+++−=+

+−

=−

K1xln2

1K1xln1xln

2

1dx

1x

1

2

1dx

1x

1

2

1dx

1x

x 22

1x

)1x(B)1x(A

1x

B

1x

A

1x

x22 −

−++=+

+−

=−

)1x(B)1x(Ax −++=

2

1AA211xSi =⇒=⇒=

2

1BB211xSi =⇒−=−⇒−=

16. ( )∫ ∫∫∫∫∫ =+

++

−+=++−+−=

+−

dx1x

12dx

1x

x

x

2xlndx

1x

2xdx

x

12dx

x

1dx

1xx

2x222222

Karctgx21xln2

1

x

2xln 2 ++−−+=

xcosv

dxeu

dxsenxdv

eu

x

x

−==

==

senxv

dxedu

dxxcosdv

eu

x

x

==

==

2

ev

dxx3du

dxedv

xu

x2

2

x2

3

=

=

=

=

2

ev

xdx2du

dxedv

xu

x2

x2

2

=

==

=

2

ev

dxdu

dxedv

xu

x2

x2

=

==

=

EJERCICIOS DE INTEGRALES INDEFINIDAS

1. ∫ dxx3senx2 = ∫ ∫−−=

−−− xdx3cosx3

2

3

x3cosxdx

3

x3cosx2

3

x3cosx 22

=

( )∗

( )∗ =

−−− ∫ dx3

x3sen

3

x3xsen

3

2

3

x3cosx2

=

++−− Kx3cos9

1

3

x3xsen

3

2

3

x3cosx2

= K27

x3cos2

9

x3xsen2

3

x3cosx2

++−−

2. ∫ +dx

9x

x2

= ∫ ++=+

K9xln2

1dx

9x

x2

2

1 22

3. ∫−

dxx1

x4

=

( ) ( )( )∫ ∫ +=

−=

−Kxarcsen

2

1dx

x1

x2

2

1dx

x1

x 2

2222

4. ( )∫ ++=+

= K1elndx1e

e)x(F x

x

x

si 2ln2K2K2ln2)0(F −=⇒=+⇒=

( ) 2ln21eln)x(F x −++=⇒

5. dx7x5

x23∫ +

−=

∫ ∫∫ =+

+−=+

+−=

++− dx

7x5

5

25

29x

5

2dx

7x5

1

5

29x

5

2dx

7x5

529

5

2

= K7x5ln25

29x

5

2 +++−

6. ( )∫ dx

x

xln 4

= ( ) Kxln5

1 5 +

7. ∫− dxex

2x3 = ∫ +−−=−−−−−

−−

K2

e

2

exdxxe2

2

1

2

ex22

2

2 xx2x

x2

3

x3cosv

xdx2du

xdx3sendv

xu 2

−=

==

=

3

x3senv

dxdu

xdx3cosdv

xu

=

==

=

2

2

x

x

2

e2

1v

xdx2du

dxxedv

xu

−

−

−=

==

=

8. ∫ dxex

1 x

1

2 = Ke x

1

+−

9. ( )∫ dxxcos = [ ]∫ ∫ ∫ ++=−== Ktcos2tsent2dtsenttsent2dttcost2dtt2tcos

=

= Kxcos2xsenx2 ++

10. ( )∫ dxxlnsenx =

∫ ∫ =−=− dx)xcos(lnx2

1

2

)x(lnsenxdx

x

1)xcos(ln

2

x

2

)x(lnsenx 222

=

+− ∫ dx

x

1)x(lnsen

2

x

2

)xcos(lnx

2

1

2

)x(lnsenx 222

=

∫−− dx)x(lnxsen4

1

4

)xcos(lnx

2

)x(lnsenx 22

I4

1

4

)xcos(lnx

2

)x(lnsenxI

22

−−=

4

)xcos(lnx

2

)x(lnsenxI

4

5 22

−=

K4

)xcos(lnx

2

)x(lnsenx

5

4dx)x(lnxsen

22

+

−=∫

11. ∫ −+

dxxx

1x2

= ∫ ∫ +−+−=−

+−K1xln2xlndx

1x

2dx

x

1

)1x(x

Bx)1x(A

1x

B

x

A

xx

1x2 −

+−=−

+=−

+

Bx)1x(A1x +−=+

2B1x

1A0x

=⇒=−=⇒=

tdt2dx

tx

tx2

==

=

sentv

dxdu

tdtcosdv

tu

===

=

2

xv

dxx

1)xcos(lndu

xdxdv

)x(lnsenu

2

=

=

==

2

xv

dxx

1)x(lnsendu

xdxdv

)xcos(lnu

2

=

−=

==

12. ∫ ∫ ∫ ∫ ∫ =++=+=+=+ − K

21

x3

25

xdxx3dxx

dxx

3dx

x

xdx

x

3x2125

212322

Kx65

xx2Kx6

5

x2 25

++=++=

13. ∫ ∫ ∫−+−=+−= dxxcosesenxexcosedxxcosexcosedxsenxe xxxxxx

IsenxexcoseI xx −+−=

senxexcoseI2 xx +−=

( )∫ ++−= Ksenxexcose2

1dxsenxe xxx

14. ∫ ∫ ∫ =−=−= dxex2

3

2

exdxx3e

2

1

2

exdxex

x22x23

2x2x23x23

=

−−= ∫ dxxe

2

ex

2

3

2

ex x2x22x23

=

−+−= ∫ dx

2

e

2

xe

2

3

4

ex3

2

ex x2x2x22x23

K8

e3

4

xe3

4

ex3

2

ex x2x2x22x23

+−+−=

15. [ ]∫ ∫ ∫ +−=+++−=+

+−

=−

K1xln2

1K1xln1xln

2

1dx

1x

1

2

1dx

1x

1

2

1dx

1x

x 22

1x

)1x(B)1x(A

1x

B

1x

A

1x

x22 −

−++=+

+−

=−

)1x(B)1x(Ax −++=

2

1AA211xSi =⇒=⇒=

2

1BB211xSi =⇒−=−⇒−=

16. ( )∫ ∫∫∫∫∫ =+

++

−+=++−+−=

+−

dx1x

12dx

1x

x

x

2xlndx

1x

2xdx

x

12dx

x

1dx

1xx

2x222222

Karctgx21xln2

1

x

2xln 2 ++−−+=

xcosv

dxeu

dxsenxdv

eu

x

x

−==

==

senxv

dxedu

dxxcosdv

eu

x

x

==

==

2

ev

dxx3du

dxedv

xu

x2

2

x2

3

=

=

=

=

2

ev

xdx2du

dxedv

xu

x2

x2

2

=

==

=

2

ev

dxdu

dxedv

xu

x2

x2

=

==

=

EJERCICIOS DE INTEGRALES INDEFINIDAS

1. ∫ dxx3senx2 = ∫ ∫−−=

−−− xdx3cosx3

2

3

x3cosxdx

3

x3cosx2

3

x3cosx 22

=

( )∗

( )∗ =

−−− ∫ dx3

x3sen

3

x3xsen

3

2

3

x3cosx2

=

++−− Kx3cos9

1

3

x3xsen

3

2

3

x3cosx2

= K27

x3cos2

9

x3xsen2

3

x3cosx2

++−−

2. ∫ +dx

9x

x2

= ∫ ++=+

K9xln2

1dx

9x

x2

2

1 22

3. ∫−

dxx1

x4

=

( ) ( )( )∫ ∫ +=

−=

−Kxarcsen

2

1dx

x1

x2

2

1dx

x1

x 2

2222

4. ( )∫ ++=+

= K1elndx1e

e)x(F x

x

x

si 2ln2K2K2ln2)0(F −=⇒=+⇒=

( ) 2ln21eln)x(F x −++=⇒

5. dx7x5

x23∫ +

−=

∫ ∫∫ =+

+−=+

+−=

++− dx

7x5

5

25

29x

5

2dx

7x5

1

5

29x

5

2dx

7x5

529

5

2

= K7x5ln25

29x

5

2 +++−

6. ( )∫ dx

x

xln 4

= ( ) Kxln5

1 5 +

7. ∫− dxex

2x3 = ∫ +−−=−−−−−

−−

K2

e

2

exdxxe2

2

1

2

ex22

2

2 xx2x

x2

3

x3cosv

xdx2du

xdx3sendv

xu 2

−=

==

=

3

x3senv

dxdu

xdx3cosdv

xu

=

==

=

2

2

x

x

2

e2

1v

xdx2du

dxxedv

xu

−

−

−=

==

=

8. ∫ dxex

1 x

1

2 = Ke x

1

+−

9. ( )∫ dxxcos = [ ]∫ ∫ ∫ ++=−== Ktcos2tsent2dtsenttsent2dttcost2dtt2tcos

=

= Kxcos2xsenx2 ++

10. ( )∫ dxxlnsenx =

∫ ∫ =−=− dx)xcos(lnx2

1

2

)x(lnsenxdx

x

1)xcos(ln

2

x

2

)x(lnsenx 222

=

+− ∫ dx

x

1)x(lnsen

2

x

2

)xcos(lnx

2

1

2

)x(lnsenx 222

=

∫−− dx)x(lnxsen4

1

4

)xcos(lnx

2

)x(lnsenx 22

I4

1

4

)xcos(lnx

2

)x(lnsenxI

22

−−=

4

)xcos(lnx

2

)x(lnsenxI

4

5 22

−=

K4

)xcos(lnx

2

)x(lnsenx

5

4dx)x(lnxsen

22

+

−=∫

11. ∫ −+

dxxx

1x2

= ∫ ∫ +−+−=−

+−K1xln2xlndx

1x

2dx

x

1

)1x(x

Bx)1x(A

1x

B

x

A

xx

1x2 −

+−=−

+=−

+

Bx)1x(A1x +−=+

2B1x

1A0x

=⇒=−=⇒=

tdt2dx

tx

tx2

==

=

sentv

dxdu

tdtcosdv

tu

===

=

2

xv

dxx

1)xcos(lndu

xdxdv

)x(lnsenu

2

=

=

==

2

xv

dxx

1)x(lnsendu

xdxdv

)xcos(lnu

2

=

−=

==

12. ∫ ∫ ∫ ∫ ∫ =++=+=+=+ − K

21

x3

25

xdxx3dxx

dxx

3dx

x

xdx

x

3x2125

212322

Kx65

xx2Kx6

5

x2 25

++=++=

13. ∫ ∫ ∫−+−=+−= dxxcosesenxexcosedxxcosexcosedxsenxe xxxxxx

IsenxexcoseI xx −+−=

senxexcoseI2 xx +−=

( )∫ ++−= Ksenxexcose2

1dxsenxe xxx

14. ∫ ∫ ∫ =−=−= dxex2

3

2

exdxx3e

2

1

2

exdxex

x22x23

2x2x23x23

=

−−= ∫ dxxe

2

ex

2

3

2

ex x2x22x23

=

−+−= ∫ dx

2

e

2

xe

2

3

4

ex3

2

ex x2x2x22x23

K8

e3

4

xe3

4

ex3

2

ex x2x2x22x23

+−+−=

15. [ ]∫ ∫ ∫ +−=+++−=+

+−

=−

K1xln2

1K1xln1xln

2

1dx

1x

1

2

1dx

1x

1

2

1dx

1x

x 22

1x

)1x(B)1x(A

1x

B

1x

A

1x

x22 −

−++=+

+−

=−

)1x(B)1x(Ax −++=

2

1AA211xSi =⇒=⇒=

2

1BB211xSi =⇒−=−⇒−=

16. ( )∫ ∫∫∫∫∫ =+

++

−+=++−+−=

+−

dx1x

12dx

1x

x

x

2xlndx

1x

2xdx

x

12dx

x

1dx

1xx

2x222222

Karctgx21xln2

1

x

2xln 2 ++−−+=

xcosv

dxeu

dxsenxdv

eu

x

x

−==

==

senxv

dxedu

dxxcosdv

eu

x

x

==

==

2

ev

dxx3du

dxedv

xu

x2

2

x2

3

=

=

=

=

2

ev

xdx2du

dxedv

xu

x2

x2

2

=

==

=

2

ev

dxdu

dxedv

xu

x2

x2

=

==

=

EJERCICIOS DE INTEGRALES INDEFINIDAS

1. ∫ dxx3senx2 = ∫ ∫−−=

−−− xdx3cosx3

2

3

x3cosxdx

3

x3cosx2

3

x3cosx 22

=

( )∗

( )∗ =

−−− ∫ dx3

x3sen

3

x3xsen

3

2

3

x3cosx2

=

++−− Kx3cos9

1

3

x3xsen

3

2

3

x3cosx2

= K27

x3cos2

9

x3xsen2

3

x3cosx2

++−−

2. ∫ +dx

9x

x2

= ∫ ++=+

K9xln2

1dx

9x

x2

2

1 22

3. ∫−

dxx1

x4

=

( ) ( )( )∫ ∫ +=

−=

−Kxarcsen

2

1dx

x1

x2

2

1dx

x1

x 2

2222

4. ( )∫ ++=+

= K1elndx1e

e)x(F x

x

x

si 2ln2K2K2ln2)0(F −=⇒=+⇒=

( ) 2ln21eln)x(F x −++=⇒

5. dx7x5

x23∫ +

−=

∫ ∫∫ =+

+−=+

+−=

++− dx

7x5

5

25

29x

5

2dx

7x5

1

5

29x

5

2dx

7x5

529

5

2

= K7x5ln25

29x

5

2 +++−

6. ( )∫ dx

x

xln 4

= ( ) Kxln5

1 5 +

7. ∫− dxex

2x3 = ∫ +−−=−−−−−

−−

K2

e

2

exdxxe2

2

1

2

ex22

2

2 xx2x

x2

3

x3cosv

xdx2du

xdx3sendv

xu 2

−=

==

=

3

x3senv

dxdu

xdx3cosdv

xu

=

==

=

2

2

x

x

2

e2

1v

xdx2du

dxxedv

xu

−

−

−=

==

=

8. ∫ dxex

1 x

1

2 = Ke x

1

+−

9. ( )∫ dxxcos = [ ]∫ ∫ ∫ ++=−== Ktcos2tsent2dtsenttsent2dttcost2dtt2tcos

=

= Kxcos2xsenx2 ++

10. ( )∫ dxxlnsenx =

∫ ∫ =−=− dx)xcos(lnx2

1

2

)x(lnsenxdx

x

1)xcos(ln

2

x

2

)x(lnsenx 222

=

+− ∫ dx

x

1)x(lnsen

2

x

2

)xcos(lnx

2

1

2

)x(lnsenx 222

=

∫−− dx)x(lnxsen4

1

4

)xcos(lnx

2

)x(lnsenx 22

I4

1

4

)xcos(lnx

2

)x(lnsenxI

22

−−=

4

)xcos(lnx

2

)x(lnsenxI

4

5 22

−=

K4

)xcos(lnx

2

)x(lnsenx

5

4dx)x(lnxsen

22

+

−=∫

11. ∫ −+

dxxx

1x2

= ∫ ∫ +−+−=−

+−K1xln2xlndx

1x

2dx

x

1

)1x(x

Bx)1x(A

1x

B

x

A

xx

1x2 −

+−=−

+=−

+

Bx)1x(A1x +−=+

2B1x

1A0x

=⇒=−=⇒=

tdt2dx

tx

tx2

==

=

sentv

dxdu

tdtcosdv

tu

===

=

2

xv

dxx

1)xcos(lndu

xdxdv

)x(lnsenu

2

=

=

==

2

xv

dxx

1)x(lnsendu

xdxdv

)xcos(lnu

2

=

−=

==

12. ∫ ∫ ∫ ∫ ∫ =++=+=+=+ − K

21

x3

25

xdxx3dxx

dxx

3dx

x

xdx

x

3x2125

212322

Kx65

xx2Kx6

5

x2 25

++=++=

13. ∫ ∫ ∫−+−=+−= dxxcosesenxexcosedxxcosexcosedxsenxe xxxxxx

IsenxexcoseI xx −+−=

senxexcoseI2 xx +−=

( )∫ ++−= Ksenxexcose2

1dxsenxe xxx

14. ∫ ∫ ∫ =−=−= dxex2

3

2

exdxx3e

2

1

2

exdxex

x22x23

2x2x23x23

=

−−= ∫ dxxe

2

ex

2

3

2

ex x2x22x23

=

−+−= ∫ dx

2

e

2

xe

2

3

4

ex3

2

ex x2x2x22x23

K8

e3

4

xe3

4

ex3

2

ex x2x2x22x23

+−+−=

15. [ ]∫ ∫ ∫ +−=+++−=+

+−

=−

K1xln2

1K1xln1xln

2

1dx

1x

1

2

1dx

1x

1

2

1dx

1x

x 22

1x

)1x(B)1x(A

1x

B

1x

A

1x

x22 −

−++=+

+−

=−

)1x(B)1x(Ax −++=

2

1AA211xSi =⇒=⇒=

2

1BB211xSi =⇒−=−⇒−=

16. ( )∫ ∫∫∫∫∫ =+

++

−+=++−+−=

+−

dx1x

12dx

1x

x

x

2xlndx

1x

2xdx

x

12dx

x

1dx

1xx

2x222222

Karctgx21xln2

1

x

2xln 2 ++−−+=

xcosv

dxeu

dxsenxdv

eu

x

x

−==

==

senxv

dxedu

dxxcosdv

eu

x

x

==

==

2

ev

dxx3du

dxedv

xu

x2

2

x2

3

=

=

=

=

2

ev

xdx2du

dxedv

xu

x2

x2

2

=

==

=

2

ev

dxdu

dxedv

xu

x2

x2

=

==

=

EJERCICIOS DE INTEGRALES INDEFINIDAS

1. ∫ dxx3senx2 = ∫ ∫−−=

−−− xdx3cosx3

2

3

x3cosxdx

3

x3cosx2

3

x3cosx 22

=

( )∗

( )∗ =

−−− ∫ dx3

x3sen

3

x3xsen

3

2

3

x3cosx2

=

++−− Kx3cos9

1

3

x3xsen

3

2

3

x3cosx2

= K27

x3cos2

9

x3xsen2

3

x3cosx2

++−−

2. ∫ +dx

9x

x2

= ∫ ++=+

K9xln2

1dx

9x

x2

2

1 22

3. ∫−

dxx1

x4

=

( ) ( )( )∫ ∫ +=

−=

−Kxarcsen

2

1dx

x1

x2

2

1dx

x1

x 2

2222

4. ( )∫ ++=+

= K1elndx1e

e)x(F x

x

x

si 2ln2K2K2ln2)0(F −=⇒=+⇒=

( ) 2ln21eln)x(F x −++=⇒

5. dx7x5

x23∫ +

−=

∫ ∫∫ =+

+−=+

+−=

++− dx

7x5

5

25

29x

5

2dx

7x5

1

5

29x

5

2dx

7x5

529

5

2

= K7x5ln25

29x

5

2 +++−

6. ( )∫ dx

x

xln 4

= ( ) Kxln5

1 5 +

7. ∫− dxex

2x3 = ∫ +−−=−−−−−

−−

K2

e

2

exdxxe2

2

1

2

ex22

2

2 xx2x

x2

3

x3cosv

xdx2du

xdx3sendv

xu 2

−=

==

=

3

x3senv

dxdu

xdx3cosdv

xu

=

==

=

2

2

x

x

2

e2

1v

xdx2du

dxxedv

xu

−

−

−=

==

=

8. ∫ dxex

1 x

1

2 = Ke x

1

+−

9. ( )∫ dxxcos = [ ]∫ ∫ ∫ ++=−== Ktcos2tsent2dtsenttsent2dttcost2dtt2tcos

=

= Kxcos2xsenx2 ++

10. ( )∫ dxxlnsenx =

∫ ∫ =−=− dx)xcos(lnx2

1

2

)x(lnsenxdx

x

1)xcos(ln

2

x

2

)x(lnsenx 222

=

+− ∫ dx

x

1)x(lnsen

2

x

2

)xcos(lnx

2

1

2

)x(lnsenx 222

=

∫−− dx)x(lnxsen4

1

4

)xcos(lnx

2

)x(lnsenx 22

I4

1

4

)xcos(lnx

2

)x(lnsenxI

22

−−=

4

)xcos(lnx

2

)x(lnsenxI

4

5 22

−=

K4

)xcos(lnx

2

)x(lnsenx

5

4dx)x(lnxsen

22

+

−=∫

11. ∫ −+

dxxx

1x2

= ∫ ∫ +−+−=−

+−K1xln2xlndx

1x

2dx

x

1

)1x(x

Bx)1x(A

1x

B

x

A

xx

1x2 −

+−=−

+=−

+

Bx)1x(A1x +−=+

2B1x

1A0x

=⇒=−=⇒=

tdt2dx

tx

tx2

==

=

sentv

dxdu

tdtcosdv

tu

===

=

2

xv

dxx

1)xcos(lndu

xdxdv

)x(lnsenu

2

=

=

==

2

xv

dxx

1)x(lnsendu

xdxdv

)xcos(lnu

2

=

−=

==

12. ∫ ∫ ∫ ∫ ∫ =++=+=+=+ − K

21

x3

25

xdxx3dxx

dxx

3dx

x

xdx

x

3x2125

212322

Kx65

xx2Kx6

5

x2 25

++=++=

13. ∫ ∫ ∫−+−=+−= dxxcosesenxexcosedxxcosexcosedxsenxe xxxxxx

IsenxexcoseI xx −+−=

senxexcoseI2 xx +−=

( )∫ ++−= Ksenxexcose2

1dxsenxe xxx

14. ∫ ∫ ∫ =−=−= dxex2

3

2

exdxx3e

2

1

2

exdxex

x22x23

2x2x23x23

=

−−= ∫ dxxe

2

ex

2

3

2

ex x2x22x23

=

−+−= ∫ dx

2

e

2

xe

2

3

4

ex3

2

ex x2x2x22x23

K8

e3

4

xe3

4

ex3

2

ex x2x2x22x23

+−+−=

15. [ ]∫ ∫ ∫ +−=+++−=+

+−

=−

K1xln2

1K1xln1xln

2

1dx

1x

1

2

1dx

1x

1

2

1dx

1x

x 22

1x

)1x(B)1x(A

1x

B

1x

A

1x

x22 −

−++=+

+−

=−

)1x(B)1x(Ax −++=

2

1AA211xSi =⇒=⇒=

2

1BB211xSi =⇒−=−⇒−=

16. ( )∫ ∫∫∫∫∫ =+

++

−+=++−+−=

+−

dx1x

12dx

1x

x

x

2xlndx

1x

2xdx

x

12dx

x

1dx

1xx

2x222222

Karctgx21xln2

1

x

2xln 2 ++−−+=

xcosv

dxeu

dxsenxdv

eu

x

x

−==

==

senxv

dxedu

dxxcosdv

eu

x

x

==

==

2

ev

dxx3du

dxedv

xu

x2

2

x2

3

=

=

=

=

2

ev

xdx2du

dxedv

xu

x2

x2

2

=

==

=

2

ev

dxdu

dxedv

xu

x2

x2

=

==

=

EJERCICIOS DE INTEGRALES INDEFINIDAS

1. ∫ dxx3senx2 = ∫ ∫−−=

−−− xdx3cosx3

2

3

x3cosxdx

3

x3cosx2

3

x3cosx 22

=

( )∗

( )∗ =

−−− ∫ dx3

x3sen

3

x3xsen

3

2

3

x3cosx2

=

++−− Kx3cos9

1

3

x3xsen

3

2

3

x3cosx2

= K27

x3cos2

9

x3xsen2

3

x3cosx2

++−−

2. ∫ +dx

9x

x2

= ∫ ++=+

K9xln2

1dx

9x

x2

2

1 22

3. ∫−

dxx1

x4

=

( ) ( )( )∫ ∫ +=

−=

−Kxarcsen

2

1dx

x1

x2

2

1dx

x1

x 2

2222

4. ( )∫ ++=+

= K1elndx1e

e)x(F x

x

x

si 2ln2K2K2ln2)0(F −=⇒=+⇒=

( ) 2ln21eln)x(F x −++=⇒

5. dx7x5

x23∫ +

−=

∫ ∫∫ =+

+−=+

+−=

++− dx

7x5

5

25

29x

5

2dx

7x5

1

5

29x

5

2dx

7x5

529

5

2

= K7x5ln25

29x

5

2 +++−

6. ( )∫ dx

x

xln 4

= ( ) Kxln5

1 5 +

7. ∫− dxex

2x3 = ∫ +−−=−−−−−

−−

K2

e

2

exdxxe2

2

1

2

ex22

2

2 xx2x

x2

3

x3cosv

xdx2du

xdx3sendv

xu 2

−=

==

=

3

x3senv

dxdu

xdx3cosdv

xu

=

==

=

2

2

x

x

2

e2

1v

xdx2du

dxxedv

xu

−

−

−=

==

=

8. ∫ dxex

1 x

1

2 = Ke x

1

+−

9. ( )∫ dxxcos = [ ]∫ ∫ ∫ ++=−== Ktcos2tsent2dtsenttsent2dttcost2dtt2tcos

=

= Kxcos2xsenx2 ++

10. ( )∫ dxxlnsenx =

∫ ∫ =−=− dx)xcos(lnx2

1

2

)x(lnsenxdx

x

1)xcos(ln

2

x

2

)x(lnsenx 222

=

+− ∫ dx

x

1)x(lnsen

2

x

2

)xcos(lnx

2

1

2

)x(lnsenx 222

=

∫−− dx)x(lnxsen4

1

4

)xcos(lnx

2

)x(lnsenx 22

I4

1

4

)xcos(lnx

2

)x(lnsenxI

22

−−=

4

)xcos(lnx

2

)x(lnsenxI

4

5 22

−=

K4

)xcos(lnx

2

)x(lnsenx

5

4dx)x(lnxsen

22

+

−=∫

11. ∫ −+

dxxx

1x2

= ∫ ∫ +−+−=−

+−K1xln2xlndx

1x

2dx

x

1

)1x(x

Bx)1x(A

1x

B

x

A

xx

1x2 −

+−=−

+=−

+

Bx)1x(A1x +−=+

2B1x

1A0x

=⇒=−=⇒=

tdt2dx

tx

tx2

==

=

sentv

dxdu

tdtcosdv

tu

===

=

2

xv

dxx

1)xcos(lndu

xdxdv

)x(lnsenu

2

=

=

==

2

xv

dxx

1)x(lnsendu

xdxdv

)xcos(lnu

2

=

−=

==

12. ∫ ∫ ∫ ∫ ∫ =++=+=+=+ − K

21

x3

25

xdxx3dxx

dxx

3dx

x

xdx

x

3x2125

212322

Kx65

xx2Kx6

5

x2 25

++=++=

13. ∫ ∫ ∫−+−=+−= dxxcosesenxexcosedxxcosexcosedxsenxe xxxxxx

IsenxexcoseI xx −+−=

senxexcoseI2 xx +−=

( )∫ ++−= Ksenxexcose2

1dxsenxe xxx

14. ∫ ∫ ∫ =−=−= dxex2

3

2

exdxx3e

2

1

2

exdxex

x22x23

2x2x23x23

=

−−= ∫ dxxe

2

ex

2

3

2

ex x2x22x23

=

−+−= ∫ dx

2

e

2

xe

2

3

4

ex3

2

ex x2x2x22x23

K8

e3

4

xe3

4

ex3

2

ex x2x2x22x23

+−+−=

15. [ ]∫ ∫ ∫ +−=+++−=+

+−

=−

K1xln2

1K1xln1xln

2

1dx

1x

1

2

1dx

1x

1

2

1dx

1x

x 22

1x

)1x(B)1x(A

1x

B

1x

A

1x

x22 −

−++=+

+−

=−

)1x(B)1x(Ax −++=

2

1AA211xSi =⇒=⇒=

2

1BB211xSi =⇒−=−⇒−=

16. ( )∫ ∫∫∫∫∫ =+

++

−+=++−+−=

+−

dx1x

12dx

1x

x

x

2xlndx

1x

2xdx

x

12dx

x

1dx

1xx

2x222222

Karctgx21xln2

1

x

2xln 2 ++−−+=

xcosv

dxeu

dxsenxdv

eu

x

x

−==

==

senxv

dxedu

dxxcosdv

eu

x

x

==

==

2

ev

dxx3du

dxedv

xu

x2

2

x2

3

=

=

=

=

2

ev

xdx2du

dxedv

xu

x2

x2

2

=

==

=

2

ev

dxdu

dxedv

xu

x2

x2

=

==

=

EJERCICIOS DE INTEGRALES INDEFINIDAS

1. ∫ dxx3senx2 = ∫ ∫−−=

−−− xdx3cosx3

2

3

x3cosxdx

3

x3cosx2

3

x3cosx 22

=

( )∗

( )∗ =

−−− ∫ dx3

x3sen

3

x3xsen

3

2

3

x3cosx2

=

++−− Kx3cos9

1

3

x3xsen

3

2

3

x3cosx2

= K27

x3cos2

9

x3xsen2

3

x3cosx2

++−−

2. ∫ +dx

9x

x2

= ∫ ++=+

K9xln2

1dx

9x

x2

2

1 22

3. ∫−

dxx1

x4

=

( ) ( )( )∫ ∫ +=

−=

−Kxarcsen

2

1dx

x1

x2

2

1dx

x1

x 2

2222

4. ( )∫ ++=+

= K1elndx1e

e)x(F x

x

x

si 2ln2K2K2ln2)0(F −=⇒=+⇒=

( ) 2ln21eln)x(F x −++=⇒

5. dx7x5

x23∫ +

−=

∫ ∫∫ =+

+−=+

+−=

++− dx

7x5

5

25

29x

5

2dx

7x5

1

5

29x

5

2dx

7x5

529

5

2

= K7x5ln25

29x

5

2 +++−

6. ( )∫ dx

x

xln 4

= ( ) Kxln5

1 5 +

7. ∫− dxex

2x3 = ∫ +−−=−−−−−

−−

K2

e

2

exdxxe2

2

1

2

ex22

2

2 xx2x

x2

3

x3cosv

xdx2du

xdx3sendv

xu 2

−=

==

=

3

x3senv

dxdu

xdx3cosdv

xu

=

==

=

2

2

x

x

2

e2

1v

xdx2du

dxxedv

xu

−

−

−=

==

=

8. ∫ dxex

1 x

1

2 = Ke x

1

+−

9. ( )∫ dxxcos = [ ]∫ ∫ ∫ ++=−== Ktcos2tsent2dtsenttsent2dttcost2dtt2tcos

=

= Kxcos2xsenx2 ++

10. ( )∫ dxxlnsenx =

∫ ∫ =−=− dx)xcos(lnx2

1

2

)x(lnsenxdx

x

1)xcos(ln

2

x

2

)x(lnsenx 222

=

+− ∫ dx

x

1)x(lnsen

2

x

2

)xcos(lnx

2

1

2

)x(lnsenx 222

=

∫−− dx)x(lnxsen4

1

4

)xcos(lnx

2

)x(lnsenx 22

I4

1

4

)xcos(lnx

2

)x(lnsenxI

22

−−=

4

)xcos(lnx

2

)x(lnsenxI

4

5 22

−=

K4

)xcos(lnx

2

)x(lnsenx

5

4dx)x(lnxsen

22

+

−=∫

11. ∫ −+

dxxx

1x2

= ∫ ∫ +−+−=−

+−K1xln2xlndx

1x

2dx

x

1

)1x(x

Bx)1x(A

1x

B

x

A

xx

1x2 −

+−=−

+=−

+

Bx)1x(A1x +−=+

2B1x

1A0x

=⇒=−=⇒=

tdt2dx

tx

tx2

==

=

sentv

dxdu

tdtcosdv

tu

===

=

2

xv

dxx

1)xcos(lndu

xdxdv

)x(lnsenu

2

=

=

==

2

xv

dxx

1)x(lnsendu

xdxdv

)xcos(lnu

2

=

−=

==

12. ∫ ∫ ∫ ∫ ∫ =++=+=+=+ − K

21

x3

25

xdxx3dxx

dxx

3dx

x

xdx

x

3x2125

212322

Kx65

xx2Kx6

5

x2 25

++=++=

13. ∫ ∫ ∫−+−=+−= dxxcosesenxexcosedxxcosexcosedxsenxe xxxxxx

IsenxexcoseI xx −+−=

senxexcoseI2 xx +−=

( )∫ ++−= Ksenxexcose2

1dxsenxe xxx

14. ∫ ∫ ∫ =−=−= dxex2

3

2

exdxx3e

2

1

2

exdxex

x22x23

2x2x23x23

=

−−= ∫ dxxe

2

ex

2

3

2

ex x2x22x23

=

−+−= ∫ dx

2

e

2

xe

2

3

4

ex3

2

ex x2x2x22x23

K8

e3

4

xe3

4

ex3

2

ex x2x2x22x23

+−+−=

15. [ ]∫ ∫ ∫ +−=+++−=+

+−

=−

K1xln2

1K1xln1xln

2

1dx

1x

1

2

1dx

1x

1

2

1dx

1x

x 22

1x

)1x(B)1x(A

1x

B

1x

A

1x

x22 −

−++=+

+−

=−

)1x(B)1x(Ax −++=

2

1AA211xSi =⇒=⇒=

2

1BB211xSi =⇒−=−⇒−=

16. ( )∫ ∫∫∫∫∫ =+

++

−+=++−+−=

+−

dx1x

12dx

1x

x

x

2xlndx

1x

2xdx

x

12dx

x

1dx

1xx

2x222222

Karctgx21xln2

1

x

2xln 2 ++−−+=

xcosv

dxeu

dxsenxdv

eu

x

x

−==

==

senxv

dxedu

dxxcosdv

eu

x

x

==

==

2

ev

dxx3du

dxedv

xu

x2

2

x2

3

=

=

=

=

2

ev

xdx2du

dxedv

xu

x2

x2

2

=

==

=

2

ev

dxdu

dxedv

xu

x2

x2

=

==

=

EJERCICIOS DE INTEGRALES INDEFINIDAS

1. ∫ dxx3senx2 = ∫ ∫−−=

−−− xdx3cosx3

2

3

x3cosxdx

3

x3cosx2

3

x3cosx 22

=

( )∗

( )∗ =

−−− ∫ dx3

x3sen

3

x3xsen

3

2

3

x3cosx2

=

++−− Kx3cos9

1

3

x3xsen

3

2

3

x3cosx2

= K27

x3cos2

9

x3xsen2

3

x3cosx2

++−−

2. ∫ +dx

9x

x2

= ∫ ++=+

K9xln2

1dx

9x

x2

2

1 22

3. ∫−

dxx1

x4

=

( ) ( )( )∫ ∫ +=

−=

−Kxarcsen

2

1dx

x1

x2

2

1dx

x1

x 2

2222

4. ( )∫ ++=+

= K1elndx1e

e)x(F x

x

x

si 2ln2K2K2ln2)0(F −=⇒=+⇒=

( ) 2ln21eln)x(F x −++=⇒

5. dx7x5

x23∫ +

−=

∫ ∫∫ =+

+−=+

+−=

++− dx

7x5

5

25

29x

5

2dx

7x5

1

5

29x

5

2dx

7x5

529

5

2

= K7x5ln25

29x

5

2 +++−

6. ( )∫ dx

x

xln 4

= ( ) Kxln5

1 5 +

7. ∫− dxex

2x3 = ∫ +−−=−−−−−

−−

K2

e

2

exdxxe2

2

1

2

ex22

2

2 xx2x

x2

3

x3cosv

xdx2du

xdx3sendv

xu 2

−=

==

=

3

x3senv

dxdu

xdx3cosdv

xu

=

==

=

2

2

x

x

2

e2

1v

xdx2du

dxxedv

xu

−

−

−=

==

=

8. ∫ dxex

1 x

1

2 = Ke x

1

+−

9. ( )∫ dxxcos = [ ]∫ ∫ ∫ ++=−== Ktcos2tsent2dtsenttsent2dttcost2dtt2tcos

=

= Kxcos2xsenx2 ++

10. ( )∫ dxxlnsenx =

∫ ∫ =−=− dx)xcos(lnx2

1

2

)x(lnsenxdx

x

1)xcos(ln

2

x

2

)x(lnsenx 222

=

+− ∫ dx

x

1)x(lnsen

2

x

2

)xcos(lnx

2

1

2

)x(lnsenx 222

=

∫−− dx)x(lnxsen4

1

4

)xcos(lnx

2

)x(lnsenx 22

I4

1

4

)xcos(lnx

2

)x(lnsenxI

22

−−=

4

)xcos(lnx

2

)x(lnsenxI

4

5 22

−=

K4

)xcos(lnx

2

)x(lnsenx

5

4dx)x(lnxsen

22

+

−=∫

11. ∫ −+

dxxx

1x2

= ∫ ∫ +−+−=−

+−K1xln2xlndx

1x

2dx

x

1

)1x(x

Bx)1x(A

1x

B

x

A

xx

1x2 −

+−=−

+=−

+

Bx)1x(A1x +−=+

2B1x

1A0x

=⇒=−=⇒=

tdt2dx

tx

tx2

==

=

sentv

dxdu

tdtcosdv

tu

===

=

2

xv

dxx

1)xcos(lndu

xdxdv

)x(lnsenu

2

=

=

==

2

xv

dxx

1)x(lnsendu

xdxdv

)xcos(lnu

2

=

−=

==

12. ∫ ∫ ∫ ∫ ∫ =++=+=+=+ − K

21

x3

25

xdxx3dxx

dxx

3dx

x

xdx

x

3x2125

212322

Kx65

xx2Kx6

5

x2 25

++=++=

13. ∫ ∫ ∫−+−=+−= dxxcosesenxexcosedxxcosexcosedxsenxe xxxxxx

IsenxexcoseI xx −+−=

senxexcoseI2 xx +−=

( )∫ ++−= Ksenxexcose2

1dxsenxe xxx

14. ∫ ∫ ∫ =−=−= dxex2

3

2

exdxx3e

2

1

2

exdxex

x22x23

2x2x23x23

=

−−= ∫ dxxe

2

ex

2

3

2

ex x2x22x23

=

−+−= ∫ dx

2

e

2

xe

2

3

4

ex3

2

ex x2x2x22x23

K8

e3

4

xe3

4

ex3

2

ex x2x2x22x23

+−+−=

15. [ ]∫ ∫ ∫ +−=+++−=+

+−

=−

K1xln2

1K1xln1xln

2

1dx

1x

1

2

1dx

1x

1

2

1dx

1x

x 22

1x

)1x(B)1x(A

1x

B

1x

A

1x

x22 −

−++=+

+−

=−

)1x(B)1x(Ax −++=

2

1AA211xSi =⇒=⇒=

2

1BB211xSi =⇒−=−⇒−=

16. ( )∫ ∫∫∫∫∫ =+

++

−+=++−+−=

+−

dx1x

12dx

1x

x

x

2xlndx

1x

2xdx

x

12dx

x

1dx

1xx

2x222222

Karctgx21xln2

1

x

2xln 2 ++−−+=

xcosv

dxeu

dxsenxdv

eu

x

x

−==

==

senxv

dxedu

dxxcosdv