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

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

238 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.