5 Steps to a 5 AP Calculus BC 2019

Definite Integrals 245

Definite Integrals Involving Odd and Even Functions

Iff is an even function, that is,f(−x)=f(x), and is continuous on [−a,a], then
∫a

−a

f(x)dx= 2

∫a

0

f(x)dx.

Iff is an odd function, that is,F(x)=−f(−x), and is continuous on [−a,a], then
∫a

−a

f(x)dx= 0.

Example 1

Evaluate

∫π/ 2

−π/ 2

cosxdx.

Sincef(x)=cosxis an even function,

∫π/ 2

−π/ 2

cosxdx= 2

∫π/ 2

0

cosxdx= 2 [sinx]π/ 02 = 2

[

sin

(

π

2

)

−sin( 0 )

]

= 2 ( 1 − 0 )=2.

Verify your result with a calculator.

Example 2

Evaluate

∫ 3

− 3

(

x^4 −x^2

)

dx.

Sincef(x)=x^4 −x^2 is an even function, i.e., f(−x)=f(x), thus

∫ 3

− 3

(

x^4 −x^2

)

dx= 2

∫ 3

0

(

x^4 −x^2

)

dx= 2

[

x^5

5

−

x^3

3

] 3

0

= 2

[(

35

5

−

33

3

)

− 0

]

=

396

5

.

Verify your result with a calculator.

Example 3

Evaluate

∫π

−π

sinxdx.

Sincef(x)=sinxis an odd function, i.e.,f(−x)=−f(x), thus

∫π

−π

sinxdx=0.

Verify your result algebraically.
∫π

−π

sinxdx=−cosx

]π

−π=(−cosπ)−[−cos(−π)]

=[−(− 1 )]−[−( 1 )]=( 1 )−( 1 )= 0

You can also verify the result with a calculator.