Aim: Integrate Inverse Trig Functions Course: Calculus

Do Now:

Aim: How do we integrate Inverse Trig functions?

Complete the square, express as the sum of

two squares:

x2 + 4x + 7

Aim: Integrate Inverse Trig Functions Course: Calculus

Differentiation Pairs & Integration

2

'arccos

1

d uu

dx u

2

'arctan

1

d uu

dx u

2

'arccot

1

d uu

dx u

2

'arcsec

1

d uu

dx u u

2

'arccsc

1

d uu

dx u u

2

'arcsin

1

d uu

dx u

Integration will only require one formula for each pair.

Aim: Integrate Inverse Trig Functions Course: Calculus

Integrals of Inverse Trig Functions

2 2

2 2

2 2

Let be a differentiable function of , and let > 0

1. arcsin

12. arctan

13. arcsec

u x a

du uC

aa udu u

Ca u a a

uduC

a au u a

24

dx

x arcsin2

xC

Aim: Integrate Inverse Trig Functions Course: Calculus

Model Problems

41

xdx

x

u = x2 1a

2

1

2 1

du

u

2 2arcsin

du uC

aa u

2du x dx 1

2du x dx

1arcsin

2u C 1 21

sin2

x C

Aim: Integrate Inverse Trig Functions Course: Calculus

Model Problems

22 9

dx

x u = 3x

2a 2 2

1arctan

du uC

a u a a

2 2

1 3

3 2 3

dx

x

1 1 3arctan

3 2 2

xC

24 9

dx

x x

u = 2x

3a 2 2

1 2

2 2 2 3

dx

x x

21 1arcsec

2 3 3

xC

2 2

1arcsec

uduC

a au u a

du = 2dx

du = 3dx

Aim: Integrate Inverse Trig Functions Course: Calculus

Model Problem – Not Quite

2 1x

dx

e 2 2

2 2

2 2

1. arcsin

12. arctan

13. arcsec

du uC

aa udu u

Ca u a a

uduC

a au u a

u = ex

du = exdxdx = du/ex

21x

dx

e

2

/

1

du u

u

2 1

du

u u arcsec arcsec1

xuC e C

arcsin xe C

rewrite

substitute

= du/u

substitute/rewrite and back-substitute

Aim: Integrate Inverse Trig Functions Course: Calculus

Model Problem

2

2

4

xdx

x

2 2

2 2

2 2

1. arcsin

12. arctan

13. arcsec

du uC

aa udu u

Ca u a a

uduC

a au u a

Denominator is 1 term: split integrand

2 2

2

4 4

xdx dx

x x

1 22

2

1 12 4 2

2 4x x dx dx

x

1 2241

2arcsin2 1 2 2

x xC

24 2arcsin2

xx C

Aim: Integrate Inverse Trig Functions Course: Calculus

Completing the Square

sometimes: when quad functions in integrand

2 4 7

dxdx

x x

2 2

22 2

4 7 ( 4 4) 4 7

2 3 2 3

x x x x

x x

u2 + a2

2 2

dxdx

u a

2 2

1arctan

du uC

a u a a

22 4 7 2 3

dx dxdx dx

x x x

1 2arctan

3 3

xC

Aim: Integrate Inverse Trig Functions Course: Calculus

Model Problem

2

Find the area of the region bounded by

1the graph of ( ) the -axis

33 9

and the lines = and = .2 4

f x xx x

x x

9 4

23 2

1

3dx

x x

2 2

2 2

2 2

3 33 3

2 2

3 3

2 2

x x x x

x

2 2arcsin

du uC

aa u

a = 3/2

u = x - 3/2

Complete the square

Aim: Integrate Inverse Trig Functions Course: Calculus

Model Problem

2

Find the area of the region bounded by the graph of

1 3 9 ( ) the -axis and the lines = and = .

2 43f x x x x

x x

9 4

23 2

1

3dx

x x

a = 3/2; u = x - 3/22 2

arcsindu u

Caa u

9 4

23 2 2

1

3 2 3 2x

9 4

3 2

3 2 1arcsin arcsin arcsin0

3 2 2

0.5246

x