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GII BI TP TCH PHN SUY RNG
Bi ging in t
TS. L Xun i
Trng i hc Bch Khoa TP HCM
Khoa Khoa hc ng dng, b mn Ton ng dng
Email: [email protected]
TP. HCM 2015.
TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 1 / 36
-
Tch phn suy rng loi 1 Tnh tch phn suy rng loi 1
Tnh tch phn suy rng loi 1
Cu 1
Tnh tch phn suy rng I =
+1
dx
xx2 + x + 1
.
t t =1
x x = 1
t dx = dt
t2.
i cn
x 1 +t 1 0
.
TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 2 / 36
-
Tch phn suy rng loi 1 Tnh tch phn suy rng loi 1
Tnh tch phn suy rng loi 1
Cu 1
Tnh tch phn suy rng I =
+1
dx
xx2 + x + 1
.
t t =1
x x = 1
t dx = dt
t2.
i cn
x 1 +t 1 0
.
TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 2 / 36
-
Tch phn suy rng loi 1 Tnh tch phn suy rng loi 1
Khi
I =
01
dtt2
1t .
1t2+ 1t + 1
=
10
dt1 + t + t2
=
10
d(t + 12
)(t + 12
)2+ 34
=
[ln
t + 12 +t2 + t + 1]1
0
= ln
(3
2+3
) ln
(3
2
)= ln
(1 +
23
).
TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 3 / 36
-
Tch phn suy rng loi 1 Tnh tch phn suy rng loi 1
Tnh tch phn suy rng loi 1
Cu 2
Tnh tch phn suy rng I =
+1
arctan x
x2dx .
t
u = arctan xdv = dxx2
du =
dx
1 + x2
v = 1x
TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 4 / 36
-
Tch phn suy rng loi 1 Tnh tch phn suy rng loi 1
Tnh tch phn suy rng loi 1
Cu 2
Tnh tch phn suy rng I =
+1
arctan x
x2dx .
t
u = arctan xdv = dxx2
du =
dx
1 + x2
v = 1x
TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 4 / 36
-
Tch phn suy rng loi 1 Tnh tch phn suy rng loi 1
Khi
I =
[1x. arctan x
]+1
+
+1
dx
x(1 + x2)=
=pi
4+
+1
dx
x
+1
xdx
1 + x2=pi
4+
[ln |x | 1
2ln(1 + x2)
]+1
=
=pi
4+
[ln
x1 + x2]+
1
=pi
4+ ln 1 ln 1
2=pi
4+
ln 2
2
TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 5 / 36
-
Tch phn suy rng loi 1 Tm tch phn suy rng loi 1 hi t
ngha hnh hc
Trong trng hp f (x) > 0,x [a,+), gi trca tch phn suy rng hi t c ngha hnh hc
l din tch ca hnh phng v hn c gi hn
bi x = a, trc Ox v th hm f (x)
TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 6 / 36
-
Tch phn suy rng loi 1 Tm tch phn suy rng loi 1 hi t
Ch .
T ngha hnh hc ca tch phn suy rng, ta
c nu tn ti gii hn hu hn v khc 0
limx+ f (x) = A 6= 0
v f (x) kh tch trn mi on [a, b] [a,+)th tch phn suy rng
+a
f (x)dx phn k
TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 7 / 36
-
Tch phn suy rng loi 1 Tm tch phn suy rng loi 1 hi t
1
Nu > 1 th I =+a
dx
xhi t.
2
Nu 6 1 th I =+a
dx
xphn k.
1
Nu > 1 th I =+2
dx
x. ln xhi t.
2
Nu < 1 th I =+2
dx
x. ln xphn k.
3
Nu = 1 th I =+2
dx
x. ln xhi t nu
> 1, phn k nu 6 1.
TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 8 / 36
-
Tch phn suy rng loi 1 Tm tch phn suy rng loi 1 hi t
1
Nu > 1 th I =+a
dx
xhi t.
2
Nu 6 1 th I =+a
dx
xphn k.
1
Nu > 1 th I =+2
dx
x. ln xhi t.
2
Nu < 1 th I =+2
dx
x. ln xphn k.
3
Nu = 1 th I =+2
dx
x. ln xhi t nu
> 1, phn k nu 6 1.
TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 8 / 36
-
Tch phn suy rng loi 1 Tm tch phn suy rng loi 1 hi t
1
Nu > 1 th I =+a
dx
xhi t.
2
Nu 6 1 th I =+a
dx
xphn k.
1
Nu > 1 th I =+2
dx
x. ln xhi t.
2
Nu < 1 th I =+2
dx
x. ln xphn k.
3
Nu = 1 th I =+2
dx
x. ln xhi t nu
> 1, phn k nu 6 1.
TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 8 / 36
-
Tch phn suy rng loi 1 Tm tch phn suy rng loi 1 hi t
1
Nu > 1 th I =+a
dx
xhi t.
2
Nu 6 1 th I =+a
dx
xphn k.
1
Nu > 1 th I =+2
dx
x. ln xhi t.
2
Nu < 1 th I =+2
dx
x. ln xphn k.
3
Nu = 1 th I =+2
dx
x. ln xhi t nu
> 1, phn k nu 6 1.
TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 8 / 36
-
Tch phn suy rng loi 1 Tm tch phn suy rng loi 1 hi t
1
Nu > 1 th I =+a
dx
xhi t.
2
Nu 6 1 th I =+a
dx
xphn k.
1
Nu > 1 th I =+2
dx
x. ln xhi t.
2
Nu < 1 th I =+2
dx
x. ln xphn k.
3
Nu = 1 th I =+2
dx
x. ln xhi t nu
> 1, phn k nu 6 1.TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 8 / 36
-
Tch phn suy rng loi 1 Tm tch phn suy rng loi 1 hi t
Cu 3
Tm tch phn sau hi t I =
+1
dx
x 31 + x2
1
x 31 + x2
x+ 1x.x2/3
=1
x+2/3
tch phn I hi t th +2
3> 1 > 1
3.
TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 9 / 36
-
Tch phn suy rng loi 1 Tm tch phn suy rng loi 1 hi t
Cu 3
Tm tch phn sau hi t I =
+1
dx
x 31 + x2
1
x 31 + x2
x+ 1x.x2/3
=1
x+2/3
tch phn I hi t th +2
3> 1 > 1
3.
TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 9 / 36
-
Tch phn suy rng loi 1 Tm tch phn suy rng loi 1 hi t
Cu 4
Tm tch phn sau hi t
I =
+1
(2x + 3)dx
(4 + x) 31 + x4
Trng hp 1: > 0
(2x + 3)
(4 + x) 31 + x4
x+ 2xx.x4/3
=2
x+1/3
tch phn I hi t th +1
3> 1 > 2
3.
TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 10 / 36
-
Tch phn suy rng loi 1 Tm tch phn suy rng loi 1 hi t
Cu 4
Tm tch phn sau hi t
I =
+1
(2x + 3)dx
(4 + x) 31 + x4
Trng hp 1: > 0
(2x + 3)
(4 + x) 31 + x4
x+ 2xx.x4/3
=2
x+1/3
tch phn I hi t th +1
3> 1 > 2
3.
TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 10 / 36
-
Tch phn suy rng loi 1 Tm tch phn suy rng loi 1 hi t
Trng hp 2: = 0
(2x + 3)
(4 + x) 31 + x4
x+ 2x5.x4/3
=2
5x1/3
I phn k v1
3< 1
Trng hp 3: < 0
(2x + 3)
(4 + x) 31 + x4
x+ 2x4.x4/3
=1
2x1/3
I phn k v1
3< 1.
Vy tch phn I hi t th > 2/3.
TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 11 / 36
-
Tch phn suy rng loi 1 Tm tch phn suy rng loi 1 hi t
Trng hp 2: = 0
(2x + 3)
(4 + x) 31 + x4
x+ 2x5.x4/3
=2
5x1/3
I phn k v1
3< 1
Trng hp 3: < 0
(2x + 3)
(4 + x) 31 + x4
x+ 2x4.x4/3
=1
2x1/3
I phn k v1
3< 1.
Vy tch phn I hi t th > 2/3.
TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 11 / 36
-
Tch phn suy rng loi 1 Tm tch phn suy rng loi 1 hi t
Trng hp 2: = 0
(2x + 3)
(4 + x) 31 + x4
x+ 2x5.x4/3
=2
5x1/3
I phn k v1
3< 1
Trng hp 3: < 0
(2x + 3)
(4 + x) 31 + x4
x+ 2x4.x4/3
=1
2x1/3
I phn k v1
3< 1.
Vy tch phn I hi t th > 2/3.TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 11 / 36
-
Tch phn suy rng loi 1 Tm tch phn suy rng loi 1 hi t
Cu 5
Tm tch phn sau hi t
I =
+1
(3x + 4x)dx(5 + x)1
Trng hp 1: > 0
(3x + 4x)(5 + x)1
x+ 4xx(1)
=4
x(1)1
tch phn I hi t th
( 1) 1 > 1 > 2 < 1 > 2.
TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 12 / 36
-
Tch phn suy rng loi 1 Tm tch phn suy rng loi 1 hi t
Cu 5
Tm tch phn sau hi t
I =
+1
(3x + 4x)dx(5 + x)1
Trng hp 1: > 0
(3x + 4x)(5 + x)1
x+ 4xx(1)
=4
x(1)1
tch phn I hi t th
( 1) 1 > 1 > 2 < 1 > 2.TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 12 / 36
-
Tch phn suy rng loi 1 Tm tch phn suy rng loi 1 hi t
Trng hp 2: = 0
(3x + 4x)(5 + x)1
x+ 4x61
=4
61.x1
I phn k v 1 < 1
Trng hp 3: < 0
(3x + 4x)(5 + x)1
x+ 4x51
=4
51.x1
I phn k v 1 < 1Vy tch phn I hi t th > 2.
TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 13 / 36
-
Tch phn suy rng loi 1 Tm tch phn suy rng loi 1 hi t
Trng hp 2: = 0
(3x + 4x)(5 + x)1
x+ 4x61
=4
61.x1
I phn k v 1 < 1Trng hp 3: < 0
(3x + 4x)(5 + x)1
x+ 4x51
=4
51.x1
I phn k v 1 < 1
Vy tch phn I hi t th > 2.
TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 13 / 36
-
Tch phn suy rng loi 1 Tm tch phn suy rng loi 1 hi t
Trng hp 2: = 0
(3x + 4x)(5 + x)1
x+ 4x61
=4
61.x1
I phn k v 1 < 1Trng hp 3: < 0
(3x + 4x)(5 + x)1
x+ 4x51
=4
51.x1
I phn k v 1 < 1Vy tch phn I hi t th > 2.
TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 13 / 36
-
Tch phn suy rng loi 1 Tm tch phn suy rng loi 1 hi t
Cu 6
Tm tch phn sau hi t
I =
+3
ex + ln x(1 + x)2
dx
Trng hp 1: > 0
ex + ln x(1 + x)2
x+ ln xx(2)
=1
x(2). ln1 x
tch phn I hi t th ( 2) > 1 > 1 +2 < 12 > 1 +2.
TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 14 / 36
-
Tch phn suy rng loi 1 Tm tch phn suy rng loi 1 hi t
Cu 6
Tm tch phn sau hi t
I =
+3
ex + ln x(1 + x)2
dx
Trng hp 1: > 0
ex + ln x(1 + x)2
x+ ln xx(2)
=1
x(2). ln1 x
tch phn I hi t th ( 2) > 1 > 1 +2 < 12 > 1 +2.TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 14 / 36
-
Tch phn suy rng loi 1 Tm tch phn suy rng loi 1 hi t
Trng hp 2: = 0
ex + ln x(1 + x)2
x+ ln x22
=1
22.x0 ln1 x
I phn k v 0 < 1
Trng hp 3: < 0
ex + ln x(1 + x)2
x+ ln x1
=1
x0 ln1 x
I phn k v 0 < 1
Vy tch phn I hi t th > 1 +2.
TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 15 / 36
-
Tch phn suy rng loi 1 Tm tch phn suy rng loi 1 hi t
Trng hp 2: = 0
ex + ln x(1 + x)2
x+ ln x22
=1
22.x0 ln1 x
I phn k v 0 < 1
Trng hp 3: < 0
ex + ln x(1 + x)2
x+ ln x1
=1
x0 ln1 x
I phn k v 0 < 1
Vy tch phn I hi t th > 1 +2.
TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 15 / 36
-
Tch phn suy rng loi 1 Tm tch phn suy rng loi 1 hi t
Trng hp 2: = 0
ex + ln x(1 + x)2
x+ ln x22
=1
22.x0 ln1 x
I phn k v 0 < 1
Trng hp 3: < 0
ex + ln x(1 + x)2
x+ ln x1
=1
x0 ln1 x
I phn k v 0 < 1
Vy tch phn I hi t th > 1 +2.
TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 15 / 36
-
Tch phn suy rng loi 1 Tm tch phn suy rng loi 1 hi t
Cu 7
Tm tch phn sau hi t
I =
+2
exdx
(x 1) ln x
Trng hp 1: > 0
ex
(x 1) ln xx+ e
x
x ln xx+ +
Do , tch phn I phn k.
TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 16 / 36
-
Tch phn suy rng loi 1 Tm tch phn suy rng loi 1 hi t
Cu 7
Tm tch phn sau hi t
I =
+2
exdx
(x 1) ln x
Trng hp 1: > 0
ex
(x 1) ln xx+ e
x
x ln xx+ +
Do , tch phn I phn k.
TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 16 / 36
-
Tch phn suy rng loi 1 Tm tch phn suy rng loi 1 hi t
Trng hp 2: = 0
ex
(x 1) ln xx+ 1
ln x=
1
x0. ln1 x
I phn k v 0 < 1
Trng hp 3: < 0
ex
(x 1) ln x =(x 1)ex
ln xx+ x
ex
ln x
xexln x1x2
=x+2
ex . ln xx+ 0
Vy tch phn I hi t th < 0.
TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 17 / 36
-
Tch phn suy rng loi 1 Tm tch phn suy rng loi 1 hi t
Trng hp 2: = 0
ex
(x 1) ln xx+ 1
ln x=
1
x0. ln1 x
I phn k v 0 < 1
Trng hp 3: < 0
ex
(x 1) ln x =(x 1)ex
ln xx+ x
ex
ln x
xexln x1x2
=x+2
ex . ln xx+ 0
Vy tch phn I hi t th < 0.
TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 17 / 36
-
Tch phn suy rng loi 1 Tm tch phn suy rng loi 1 hi t
Trng hp 2: = 0
ex
(x 1) ln xx+ 1
ln x=
1
x0. ln1 x
I phn k v 0 < 1
Trng hp 3: < 0
ex
(x 1) ln x =(x 1)ex
ln xx+ x
ex
ln x
xexln x1x2
=x+2
ex . ln xx+ 0
Vy tch phn I hi t th < 0.TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 17 / 36
-
Tch phn suy rng loi 2 Tnh tch phn suy rng loi 2
ngha hnh hc
Trong trng hp f (x) > 0,x [a, b), gi trca tch phn suy rng hi t c ngha hnh hc
l din tch ca hnh phng v hn c gi hn
bi x = a, x = b trc Ox v th hm f (x),
trong x = b l tim cn ng ca hm s f (x)
TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 18 / 36
-
Tch phn suy rng loi 2 Tnh tch phn suy rng loi 2
ngha hnh hc
Trong trng hp f (x) > 0,x (a, b], gi trca tch phn suy rng hi t c ngha hnh hc
l din tch ca hnh phng v hn c gi hn
bi x = a, x = b trc Ox v th hm f (x),
trong x = a l tim cn ng ca hm s f (x)
TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 19 / 36
-
Tch phn suy rng loi 2 nh ngha tch phn
ba f (x)dx, c [a, b] l im gin on
Nu hm s f (x) khng b chn khi x c, vic (a, b) th tch phn suy rng b
a
f (x)dx =
ca
f (x)dx +
bc
f (x)dx
TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 20 / 36
-
Tch phn suy rng loi 2 nh ngha tch phn
ba f (x)dx, c [a, b] l im gin on
nh l
Tch phn suy rng
ba
f (x)dx c gi l hi t
nu c 2 tch phn
ca
f (x)dx v
bc
f (x)dx u hi
t khng ph thuc ln nhau.
TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 21 / 36
-
Tch phn suy rng loi 2 nh ngha tch phn
ba f (x)dx, c [a, b] l im gin on
1
Nu < 1 th tch phnba
dx
(b x) vba
dx
(x a) hi t
2
Nu > 1 th tch phnba
dx
(b x) vba
dx
(x a) phn k
TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 22 / 36
-
Tch phn suy rng loi 2 nh ngha tch phn
ba f (x)dx, c [a, b] l im gin on
Cu 8
Tnh tch phn suy rng loi 2
I =
31
dx4x x2 3
I =
31
d(x 2)1 (x 2)2 = arcsin(x 2)
31=
= arcsin 1 arcsin(1) = pi2+pi
2= pi.
TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 23 / 36
-
Tch phn suy rng loi 2 nh ngha tch phn
ba f (x)dx, c [a, b] l im gin on
Cu 8
Tnh tch phn suy rng loi 2
I =
31
dx4x x2 3
I =
31
d(x 2)1 (x 2)2 = arcsin(x 2)
31=
= arcsin 1 arcsin(1) = pi2+pi
2= pi.
TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 23 / 36
-
Tch phn suy rng loi 2 nh ngha tch phn
ba f (x)dx, c [a, b] l im gin on
Cu 9
Tnh tch phn suy rng loi 2
I =
20
dx
(x 1)x2 x + 1
I =
10
dx
(x 1)x2 x + 1+2
1
dx
(x 1)x2 x + 1 = I1+I2
Tch phn I hi t khi c hai tch phn I1, I2 cng hi t.
TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 24 / 36
-
Tch phn suy rng loi 2 nh ngha tch phn
ba f (x)dx, c [a, b] l im gin on
Cu 9
Tnh tch phn suy rng loi 2
I =
20
dx
(x 1)x2 x + 1
I =
10
dx
(x 1)x2 x + 1+2
1
dx
(x 1)x2 x + 1 = I1+I2
Tch phn I hi t khi c hai tch phn I1, I2 cng hi t.
TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 24 / 36
-
Tch phn suy rng loi 2 nh ngha tch phn
ba f (x)dx, c [a, b] l im gin on
1
(x 1)x2 x + 1x1 1
(x 1)12 1 + 1
=1
(x 1)1Tch phn I1 phn k. Vy I cng phn k.
TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 25 / 36
-
Tch phn suy rng loi 2 Tm tch phn suy rng loi 2 hi t
Cu 10
Tm tch phn sau hi t I =
pi0
1 cos xx
dx .
1 cos xx
x0+x2
2
x=
1
2x2
tch phn I hi t th 2 < 1 < 3.
TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 26 / 36
-
Tch phn suy rng loi 2 Tm tch phn suy rng loi 2 hi t
Cu 10
Tm tch phn sau hi t I =
pi0
1 cos xx
dx .
1 cos xx
x0+x2
2
x=
1
2x2
tch phn I hi t th 2 < 1 < 3.
TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 26 / 36
-
Tch phn suy rng loi 2 Tm tch phn suy rng loi 2 hi t
Cu 11
Tm tch phn sau hi t
I =
10
e2 + x2 ecos x
xdx .
e2 + x2 ecos x
xx0+
e[(
1 + 12 .x2
e2
)(1 x22
)]x
=1 + e2
2e.
1
x2
tch phn I hi t th 2 < 1 < 3.
TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 27 / 36
-
Tch phn suy rng loi 2 Tm tch phn suy rng loi 2 hi t
Cu 11
Tm tch phn sau hi t
I =
10
e2 + x2 ecos x
xdx .
e2 + x2 ecos x
xx0+
e[(
1 + 12 .x2
e2
)(1 x22
)]x
=1 + e2
2e.
1
x2
tch phn I hi t th 2 < 1 < 3.TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 27 / 36
-
Tch phn suy rng loi 2 Tm tch phn suy rng loi 2 hi t
Cu 12
Tm tch phn sau hi t
I =
20
xdx3
(x 1)(x 2)2 .
TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 28 / 36
-
Tch phn suy rng loi 2 Tm tch phn suy rng loi 2 hi t
Trng hp 1: > 0
I =
10
xdx3
(x 1)(x 2)2+3/21
xdx3
(x 1)(x 2)2+
+
23/2
xdx3
(x 1)(x 2)2 .
TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 29 / 36
-
Tch phn suy rng loi 2 Tm tch phn suy rng loi 2 hi t
x
3
(x 1)(x 2)2x1+ 1
(x 1)1/3x
3
(x 1)(x 2)2x1 1
(x 1)1/3x
3
(x 1)(x 2)2x2 2
(x 2)2/3Do , tch phn I hi t.
TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 30 / 36
-
Tch phn suy rng loi 2 Tm tch phn suy rng loi 2 hi t
Trng hp 2: = 0
I =
10
dx3
(x 1)(x 2)2+3/21
dx3
(x 1)(x 2)2+
+
23/2
dx3
(x 1)(x 2)2 .
TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 31 / 36
-
Tch phn suy rng loi 2 Tm tch phn suy rng loi 2 hi t
13
(x 1)(x 2)2x1+ 1
(x 1)1/31
3
(x 1)(x 2)2x1 1
(x 1)1/31
3
(x 1)(x 2)2x2 1
(x 2)2/3Do , tch phn I hi t.
TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 32 / 36
-
Tch phn suy rng loi 2 Tm tch phn suy rng loi 2 hi t
Trng hp 3: < 0
I =
1/20
dx
x 3(x 1)(x 2)2 +
11/2
dx
x 3(x 1)(x 2)2+
+
3/21
dx
x 3(x 1)(x 2)2 +
23/2
dx
x 3(x 1)(x 2)2 .
TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 33 / 36
-
Tch phn suy rng loi 2 Tm tch phn suy rng loi 2 hi t
1
x 3
(1 x)(x 2)2x0+ 1
34.x
1
x 3
(x 1)(x 2)2x1+ 1
(x 1)1/31
x 3
(x 1)(x 2)2x1 1
(x 1)1/31
x 3
(x 1)(x 2)2x2 2
(x 2)2/3Do , tch phn I hi t khi < 1 > 1.TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 34 / 36
-
Tch phn suy rng loi 2 Tm tch phn suy rng loi 2 hi t
Vy tch phn I hi t th{ > 0
1 < < 0 > 1.
TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 35 / 36
-
Tch phn suy rng loi 2 Tm tch phn suy rng loi 2 hi t
THANK YOU FOR ATTENTION
TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 36 / 36
Tch phn suy rng loi 1Tnh tch phn suy rng loi 1Tm tch phn suy rng loi 1 hi t
Tch phn suy rng loi 2Tnh tch phn suy rng loi 2nh ngha tch phn ab f(x)dx, c [a,b] l im gin onTm tch phn suy rng loi 2 hi t
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