Integration 217
Step 3. Rewrite:
∫
√^1
9 −u^2
du
2
=
1
2
∫
√du
32 −u^2
.
Step 4. Integrate:
1
2
sin−^1
(
u
3
)
+C.
Step 5. Replaceu:
1
2
sin−^1
(
2 x
3
)
+C.
Step 6. Differentiate and Check:
1
2
1
√
1 −( 2 x/ 3 )^2
·
2
3
=
1
3
1
√
1 − 4 x^2 / 9
=
1
√
9
1
√
1 − 4 x^2 / 9
=
1
√
9 ( 1 − 4 x^2 / 9 )
=
1
√
9 − 4 x^2
.
Example 2
Evaluate
∫
1
x^2 + 2 x+ 5
dx.
Step 1. Rewrite:
∫
1
(x^2 + 2 x+ 1 )+ 4
=
∫
1
(x+ 1 )^2 + 22
dx
=
∫
1
22 +(x+ 1 )^2
dx.
Letu=x+1.
Step 2. Differentiate:du=dx.
Step 3. Rewrite:
∫
1
22 +u^2
du.
Step 4. Integrate:
1
2
tan−^1
(
u
2
)
+C.
Step 5. Replaceu:
1
2
tan−^1
(
x+ 1
2
)
+C.
Step 6. Differentiate and Check:
(
1
2
)
1 ( 1 / 2 )
1 +[(x+1)/ 2 ]^2
=
(
1
4
)
1
1 +(x+ 1 )^2 / 4
=
(
1
4
)
4
4 +(x+ 1 )^2
=
1
x^2 + 2 x+ 5
.
TIP • If the problem gives you that the diameter of a sphere is 6 and you are using formulas
such asv=
4
3
πr^3 ors= 4 πr^2 , do not forget thatr=3.