MA 3972-MA-Book May 8, 2018 13:52
Integration 239Step 3: Rewrite: 4
∫
1
u
du.Step 4: Integrate: 4 ln|u|+C.
Step 5: Replaceu:4ln|x− 1 |+C.
∫
x^2 + 3
x− 1
dx=
x^2
2
+x+4ln|x− 1 |+C.
Step 6: Differentiate and Check:
2 x
2+ 1 + 4
(
1
x− 1)
+C=x+ 1 +4
x− 1=
x^2 + 3
x− 1.
Example 4
Evaluate
∫
lnx
3 x
dx.Step 1: Letu=lnx.
Step 2: Differentiate:du=
1
x
dx.Step 3: Rewrite:
1
3
∫
udx.Step 4: Integrate:
(
1
3)
u^2
2+C=
1
6
u^2 +C.Step 5: Replaceu:
1
6
(lnx)^2 +C.Step 6: Differentiate and Check:
1
6
( 2 )(lnx)(
1
x)
=
lnx
3 x.
Example 5
Evaluate
∫
e(^2 x−^5 )dx.Step 1: Letu= 2 x−5.
Step 2: Differentiate:du= 2 dx⇒
du
2
=dx.
Step 3: Rewrite:
∫
eu(
du
2)
=1
2
∫
eudu.Step 4: Integrate:
1
2
eu+C.Step 5: Replaceu:
1
2
e(^2 x−^5 )+C.Step 6: Differentiate and Check:
1
2
e^2 x−^5 ( 2 )=e^2 x−^5.