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

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

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

Example 2

Evaluate

∫

3

(

sec^2 x

)√

tanxdx.

Step 1: Letu=tanx.

Step 2: Differentiate:du=sec^2 xdx.

Step 3: Rewrite: 3

∫

(tanx)^1 /^2 sec^2 xdx= 3

∫

u^1 /^2 du.

Step 4: Integrate: 3

u^3 /^2

3 / 2

+C= 2 u^3 /^2 +C.

Step 5: Replaceu: 2(tanx)^3 /^2 +Cor 2 tan^3 /^2 x+C.

Step 6: Differentiate and Check:( 2 )

(

3

2

)(

tan^1 /^2 x

)(

sec^2 x

)

= 3

(

sec^2 x

)√

tanx.

Example 3

Evaluate

∫

2 x^2 cos

(

x^3

)

dx.

Step 1: Letu=x^3.

Step 2: Differentiate:du= 3 x^2 dx⇒

du

3

=x^2 dx.

Step 3: Rewrite: 2

∫ [

cos

(

x^3

)]

x^2 dx= 2

∫

cosu

du

3

=

2

3

∫

cosudu.

Step 4: Integrate:

2

3

sinu+C.

Step 5: Replaceu:

2

3

sin

(

x^3

)

+C.

Step 6: Differentiate and Check:

2

3

[

cos

(

x^3

)]

3 x^2 = 2 x^2 cos

(

x^3

)

.

TIP • Remember that the area of a semicircle is^1

2

πr^2. Do not forget the

1

2

. If the cross

sections of a solid are semicircles the integral for the volume of the solid will involve

(

1

2

) 2

, which is

1

4

.

U-Substitution and Inverse Trigonometric Functions

Example 1

Evaluate

∫

dx

√

9 − 4 x^2

.

Step 1: Letu= 2 x.

Step 2: Differentiate:du= 2 dx;

du

2

=dx.