5 Steps to a 5 AP Calculus AB 2019 - William Ma

MA 3972-MA-Book May 8, 2018 13:52

234 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.