214 STEP 4. Review the Knowledge You Need to Score High
Example 1
Evaluate∫
x(x+1)^10 dx.Step 1. Letu=x+1; thenx=u−1.
Step 2. Differentiate:du=dx.Step 3. Rewrite:∫
(u− 1 )u^10 du=∫ (
u^11 −u^10)
du.Step 4. Integrate:
u^12
12−
u^11
11+C.
Step 5. Replaceu:
(x+ 1 )^12
12−
(x+ 1 )^11
11+C.
Step 6. Differentiate and Check:
12 (x+ 1 )^11
12−
11 (x+ 1 )^10
11
=(x+ 1 )^11 −(x+ 1 )^10
=(x+1)^10 (x+ 1 −1)
=(x+1)^10 xorx(x+1)^10.Example 2
Evaluate∫
x√
x− 2 dx.Step 1. Letu=x−2; thenx=u+ 2.
Step 2. Differentiate:du=dx.Step 3. Rewrite:∫
(u+ 2 )√
udu=∫
(u+ 2 )u^1 /^2 du=∫ (
u^3 /^2 + 2 u^1 /^2)
du.Step 4. Integrate:
u^5 /^2
5 / 2+
2 u^3 /^2
3 / 2+C.
Step 5. Replace:
2 (x− 2 )^5 /^2
5+
4 (x− 2 )^3 /^2
3+C.
Step 6. Differentiate and Check:(
5
2)
2 (x− 2 )^3 /^2
5+
(
3
2)
4 (x− 2 )^1 /^2
3
=(x− 2 )^3 /^2 + 2 (x− 2 )^1 /^2
=(x− 2 )^1 /^2 [(x−2)+ 2 ]
=(x− 2 )^1 /^2 xorx√
x−2.Example 3
Evaluate∫
(2x−5)^2 /^3 dx.Step 1. Letu= 2 x−5.
Step 2. Differentiate:du= 2 dx⇒
du
2=dx.