Integrals Related to Inverse Trig, Inverse Hyperbolic Functions

Lesson 9.5

2

Review

• Recall derivatives of inverse trig functions

1

2

12

1

2

1sin , 1

11

tan11

sec , 11

d duu u

dx dxud du

udx u dxd du

u udx dxu u

3

Integrals Using Same Relationships

2 2

2 2

2 2

arcsin

1arctan

1arcsec

du uC

aa udu u

Ca u a adu u

Ca au u a

When given integral problems,

look for these patterns

When given integral problems,

look for these patterns

4

Identifying Patterns

• For each of the integrals below, which inverse trig function is involved?

2

4

13 16

dx

x 225 4

dx

x x

29

dx

x 2 2 10

dx

x x Hint: use completing the square

Hint: use completing the square

5

Warning

• Many integrals look like the inverse trig forms

• Which of the following are of the inverse trig forms?

21

dx

x

21

x dx

x

21

dx

x

21

x dx

x

If they are not, how are they integrated?

If they are not, how are they integrated?

6

Try These

• Look for the pattern or how the expression can be manipulated into one of the patterns

2

8

1 16

dx

x

21 25

x dx

x

24 4 15

dx

x x

2

5

10 16

xdx

x x

7

Integrals Involving Inverse Hyperbolic Functions

1

2 2

1

2 2

-12 2

1

2 2

1sinh

1cosh

1 1tanh

1 1sech

udu C

au au

du Cau au

du Ca u a a

udu C

a au a u

8

Try It!

• Note the definite integral

• What is the a, the u, the du? a = 3, u = 2x, du = 2 dx

4

21

1

9 4dx

x

4

21

1 2

2 9 4dx

x

41

1

1 2sinh2 3

x

9

Application

• Find the area enclosed by x = -¼, x = ¼, y = 0, and

• Which pattern does this match?• What is the a, the u, the du?

2

1

1 4y

x

10

Assignment

• Lesson 9.5

• Page 381

• Exercises 1 – 35 odd