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

MA 3972-MA-Book April 11, 2018 15:57

Def inite Integrals 257

12.3 Evaluating Definite Integrals

Main Concepts:Definite Integrals Involving Algebraic Functions;

Definite Integrals Involving Absolute Value; Definite

Integrals Involving Trigonometric, Logarithmic, and

Exponential Functions; Definite Integrals Involving

Odd and Even Functions

TIP • If the problem asks you to determine the concavity off′(notf), you need to know

iff′′is increasing or decreasing, or iff′′′is positive or negative.

Definite Integrals Involving Algebraic Functions

Example 1

Evaluate

∫ 4

1

x^3 − 8

√

x

dx.

Rewrite:

∫ 4

1

x^3 − 8

√

x

dx=

∫ 4

1

(

x^5 /^2 − 8 x−^1 /^2

)

dx

=

x^7 /^2

7 / 2

−

8 x^1 /^2

1 / 2

] 4

1

=

2 x^7 /^2

7

− 16 x^1 /^2

] 4

1

=

(

2(4)^7 /^2

7

−16(4)^1 /^2

)

−

(

2(1)^7 /^2

7

−16(1)^1 /^2

)

=

142

7

.

Verify your result with a calculator.

Example 2

Evaluate

∫ 2

0

x(x^2 −1)^7 dx.

Begin by evaluating the indefinite integral

∫

x(x^2 −1)^7 dx.

Letu=x^2 −1;du= 2 xdxor

du

2

=xdx.

Rewrite:

∫

u^7 du

2

=

1

2

∫

u^7 du=

1

2

(

u^8

8

)

+C=

u^8

16

+C=

(x^2 −1)^8

16

+C.

Thus, the definite integral

∫ 2

0

x(x^2 −1)^7 dx=

(x^2 −1)^8

16

] 2

0

=

(2^2 −1)^8

16

−

(0^2 −1)^8

16

=

38

16

−

(−1)^8

16

=

38 − 1

16

=410.

Verify your result with a calculator.