Inverse Trigonometric Functions: Integration

Lesson 5.8

Review

• Recall derivatives of inverse trig functions

2

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

Integrals Using Same Relationships

3

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

Identifying Patterns

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

4

2

4

13 16

dx

x 225 4

dx

x x

29

dx

x

Warning

• Many integrals look like the inverse trig forms

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

5

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?

Try These

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

6

2

8

1 16

dx

x

21 25

x dx

x

24 4 15

dx

x x

2

5

10 16

xdx

x x

Completing the Square

• Often a good strategy when quadratic functions are involved in the integration

• Remember … we seek (x – b)2 + c Which might give us an integral resulting in the

arctan function

2 2 10

dx

x x

Completing the Square

• Try these2

22 4 13

dx

x x

2

2

4dx

x x

Rewriting as Sum of Two Quotients

• The integral may not appear to fit basic integration formulas May be possible to split the integrand into two

portions, each more easily handled

2

4 3

1

xdx

x

Basic Integration Rules

• Note table of basic rules Page 364

• Most of these should be committed to memory

• Note that to apply these, you must create the proper du to correspond to the u in the formula

cos sinu du u C

Assignment

• Lesson 5.8

• Page 366

• Exercises 1 – 39 odd

63, 67

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