Math 191 Lecture Notes Date §7.4 Partial Fractions 2 x +1 + 3 x - 2 Ex1: Use Partial Fractions to find the following integral Z 5x - 1 x 2 - x - 2 dx 1 LCD = ( Xt 1) ( X - 2) ×÷I÷+÷I÷ 2X-4t3×+3_ = 5×-1×2 - × - 2 XZ X - 2 = ) ×÷d× + fxtzdx = 2) d¥ , + 3) ¥2 w=x -2 dw=d× He * HA du=d× = 2 f dud + 3 fdw-w = z lnlul t 3 lnlwl t C = 2 In 1×+11+3 In 1×-21 + C ✓ = In /(xH)2( x. 2) 31 t c
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x+1 x2 ×÷I÷+÷I÷ - El Camino College Compton Center 191 Lecture Notes Date 7.4 Partial Fractions 2 x+1 + 3 x2 Ex1: Use Partial Fractions to find the following integral Z 5x1 x2
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Math 191 Lecture Notes Date
§7.4 Partial Fractions
2
x+ 1+
3
x� 2
Ex1: Use Partial Fractions to find the following integralZ
5x� 1
x2 � x� 2dx
1
LCD = ( Xt 1) ( X - 2)
×÷I÷+÷I÷2X-4t3×+3_ =5×-1×2
- × - 2 XZ . X - 2
= ) ×÷d× + fxtzdx
= 2) d¥,
+ 3) ¥2w=x -2
dw=d×He*HA
du=d×= 2 f dud + 3 fdw-w
= z lnlul t 3 lnlwl t C
= 2 In 1×+11+3In 1×-21 + C✓
= In /(xH)2(x. 2)31 t c
Ex2: Use Partial Fractions to find the following integral
Z 1
0
x� 4
x2 � 5x+ 6dx
2
•
8 X -4
¥-52 ,
/ ⇒x.FI?iIxI*IIf ×=2 ⇒ - B = - 2 ⇒ B@it X= 3 ⇒ A = - 1 ⇒ A=@
. I + S÷zd×
= -In 1×-31 t
In1×-21
= ml 't¥l+c
Ex3: Use Partial Fractions to find the following integral
Zx2 � x� 6
x2 + 3xdx
3
3. x. +3 ) as + PI =HE
X 1×2+3 )
⇒ A ( × 't 3 ) + ( Bxtc ) × = ×'
. x - 6
⇐. ⇒
axn.it?EEtEEIII.E€
ftzdx + |3×xI÷d× u=x2t3
⇒ -52¥ + 3f÷×+dy.fd×÷ ,
du=2×d×
= .zinixl + 3. tzmhi " ' '
tbta"
#g.gg#taanxa
)
Ex4: Use Partial Fractions to find the following integral
Zx3 + 6x� 2
x4 + 6x2dx
4
AXTB
0×4×2+6) f
¥
÷ + BE +' I÷e=xIII÷ ,
AXCX 2+6 ) + B ( ×2t6 ) + (C x + D) X2= X 3+6×-2
€o⇒ 613=-2 ⇒ B=@)
A X3
+ 6 Ax + 13×2+613 + C X3
+ DXZ
((1/3) ) At c= 1
( cxz ) )Bt D= 0 ⇒ D=@
(( X ) )6 A = 6 ⇒ A=@ ⇒ @
s¥s¥ts*lnlxltf.txtfozftai.lt#
Ex5: Use Partial Fractions to find the following integral
Zx5 + x� 1
x3 + 1
5
1/2×3+1115+0×4+0×3+0×2 + × - l
f. X5 + + ×2
- ( X2 . xti )=
- 0 - xztx - 1
fxzdx + f -×2¥ dx L×3tl
- ( ×2 . xtl )(
xtD(x2#)
fxzdx - fdxxtl
×z3-ln1xH@
Ex6: Use Partial Fractions to find the following integral
Zdx
x1/3 + 1
6
U = x'S
⇒ x=U3
f 2¥ ⇒ dx= 3U2 duutl
- 1
3) n÷,
du u+iu%+ou+u2+ n
=
0 - U + 0
}[ Sunda + 5¥ ] If
3 [ u÷ .
u + lulutil ] + c
0
zedo.r3xlbi.lu/x43t
Ex7: Use Partial Fractions to find the following integral