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

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

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

Example 6

Evaluate

∫

ex

ex+ 1

dx.

Step 1: Letu=ex+1.

Step 2: Differentiate:du=exdx.

Step 3: Rewrite:

∫

1

u

du.

Step 4: Integrate: ln|u|+C.

Step 5: Replaceu:ln|ex+ 1 |+C.

Step 6: Differentiate and Check:

1

ex+ 1

·ex=

ex

ex+ 1

.

Example 7

Evaluate

∫

xe^3 x^2 dx.

Step 1: Letu= 3 x^2.

Step 2: Differentiate:du= 6 xdx⇒

du

6

=xdx.

Step 3: Rewrite:

∫

eu

du

6

=

1

6

∫

eudu.

Step 4: Integrate:

1

6

eu+C.

Step 5: Replaceu:

1

6

e^3 x

2

+C.

Step 6: Differentiate and Check:

1

6

(

e^3 x^2

)

(6x)=xe^3 x^2.

Example 8

Evaluate

∫

5 (^2 x)dx.

Step 1: Letu= 2 x.

Step 2: Differentiate:du= 2 dx⇒

du

2

=dx.

Step 3: Rewrite:

∫

5 u

du

2

=

1

2

∫

5 udu.

Step 4: Integrate:

1

2

( 5 u)

ln 5

+C=

5 u

(2 ln 5)

+C.

Step 5: Replaceu:

52 x

2ln5

+C.

Step 6: Differentiate and Check:

(5^2 x)(2) ln 5

2ln5

= 52 x.