Integration of Inverse Trigonometric Functions: Review and Practice Problems, Slides of Differential and Integral Calculus

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INTEGRATION OF

INVERSE TRIGONOMETRIC

FUNCTIONS

Review

• (^) Recall derivatives of inverse trig

functions

2

1 2 1 2 1 2

1

sin , 1

1

1

tan

1

1

sec , 1

1

d du

u u

dx dx u

d du

u

dx u dx

d du

u u

dx dx u u







 







 



. If a=1, you have:



sin u C

1 u

du 1

2

 





tan u C

1 u

du 1

2





u C

u u

du 1

2

sec

Identifying Patterns

• (^) For each of the integrals below,

which inverse trig function is

involved?

5

2

dx

 x

 2

dx

x x 



2

dx

 x



Try These

• (^) Look for the pattern or how the

expression can be manipulated into

one of the patterns

7

2

dx

 x



2

1 25

x dx

 x



2

4 4 15

dx

 x  x 



2

5

10 16

x

dx

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

dx

x  x 



Example : Evaluate the following integrals using

the formulas for integrals yielding inverse

trigonometric functions:





2

2 5

s

ds





2

2

x x

dx



2

x x

dx

16  9

2

y y

dy



  1

2

x x

xdx

 2  4  3

3

2

x x x

dx



 



2 2 3

3 1

2

x x

x dx

HOMEWORK : Evaluate the following

integral.

11



4

16 9

r

rdr

 x

xdx

2

2 cos

sin

 x x

dx

1





dx

x

x

2

2

4

1



dx

e

e

x

x

2

1

 

2

3 2

x x

xdx







d 



2

1 sin

cos

2  4  3

2

2

3

x x

x dx



2

1 ln

x x

dx

9  1

2

x x

dx



1

0

2

2

t

dt

 

1

1 / 2

2

12. 3 4 x 4 x dx



ln 3

ln 2

z z

e e

dz