5 Steps to a 5 AP Calculus BC 2019

Integration 215

Step 3. Rewrite:

∫

u^2 /^3

du

2

=

1

2

∫

u^2 /^3 du.

Step 4. Integrate:

1

2

(

u^5 /^3

5 / 3

)

+C=

3 u^5 /^3

10

+C.

Step 5. Replaceu:
3 ( 2 x− 5 )^5 /^3
10

+C.

Step 6. Differentiate and Check:

(

3

10

)(

5

3

)

( 2 x− 5 )^2 /^3 (2)=( 2 x− 5 )^2 /^3.

Example 4

Evaluate

∫

x^2

(x^3 − 8 )^5

dx.

Step 1. Letu=x^3 −8.

Step 2. Differentiate:du= 3 x^2 dx⇒
du
3
=x^2 dx.

Step 3. Rewrite:

∫

1

u^5

du

3

=

1

3

∫

1

u^5

du=

1

3

∫

u−^5 du.

Step 4. Integrate:

1

3

(

u−^4

− 4

)

+C.

Step 5. Replaceu:

1

− 12

(

x^3 − 8

)− 4

+Cor

− 1

12 (x^3 − 8 )^4

+C.

Step 6. Differentiate and Check:

(

−

1

12

)

(− 4 )

(

x^3 − 8

)− 5 (

3 x^2

)

=

x^2

(x^3 − 8 )^5

.

U-Substitution and Trigonometric Functions

Example 1

Evaluate

∫

sin 4xdx.

Step 1. Letu= 4 x.

Step 2. Differentiate:du= 4 dxor
du
4
=dx.

Step 3. Rewrite:

∫

sinu

du

4

=

1

4

∫

sinudu.

Step 4. Integrate:

1

4

(−cosu)+C=−

1

4

cosu+C.

Step 5. Replaceu:−

1

4

cos( 4 x)+C.

Step 6. Differentiate and Check:

(

−

1

4

)

(−sin 4x)( 4 )=sin 4x.