5 Steps to a 5 AP Calculus BC 2019

218 STEP 4. Review the Knowledge You Need to Score High

U-Substitution and Logarithmic and Exponential Functions

Example 1

Evaluate

∫

x^3

x^4 − 1

dx.

Step 1. Letu=x^4 −1.

Step 2. Differentiate:du= 4 x^3 dx⇒

du

4

=x^3 dx.

Step 3. Rewrite:

∫

1

u

du

4

=

1

4

∫

1

u

du.

Step 4. Integrate:

1

4

ln|u|+C.

Step 5. Replaceu:

1

4

ln

∣

∣x^4 − 1

∣

∣+C.

Step 6. Differentiate and Check:

(

1

4

)

1

x^4 − 1

(

4 x^3

)

=

x^3

x^4 − 1

.

Example 2

Evaluate

∫

sinx

cosx+ 1

dx.

Step 1. Letu=cosx+1.

Step 2. Differentiate:du=−sinxdx⇒−du=sinxdx.

Step 3. Rewrite:

∫

−du

u

=−

∫

du

u

.

Step 4. Integrate:−ln|u|+C.

Step 5. Replaceu:−ln

∣∣

cosx+ 1

∣∣

+C.

Step 6. Differentiate and Check:−

(

1

cosx+ 1

)

(−sinx)=

sinx

cosx+ 1

.

Example 3

Evaluate

∫

x^2 + 3

x− 1

dx.

Step 1. Rewrite:

x^2 + 3

x− 1

=x+ 1 +

4

x− 1

by dividing (x^2 + 3) by (x−1).

∫

x^2 + 3

x− 1

dx=

∫ (

x+ 1 +

4

x− 1

)

dx=

∫

(x+ 1 )dx+

∫

4

x− 1

dx

=

x^2

2

+x+ 4

∫

1

x− 1

dx

Letu=x−1.

Step 2. Differentiate:du=dx.