1.3 Integration of inverse Trigonometric functions

MOOC MAT438/ UiTM

Inverse Trigonometric Functions

 Introduction (notation, graph and useful formula)
 Evaluating Inverse Trigonometric Functions using Triangle Method
 Derivatives of Inverse Trigonometric Functions
 Integration of Inverse Trigonometric Functions

MOOC MAT438/ UiTM

Formula (in general) Formula (in Appendix)

 න 1 = −1 +
2 − 2

 න 2 1 = 1 −1 +
+ 2

 න 1 = 1 −1 +
2 − 2

=
=

MOOC MAT438/ UiTM

න 1

3 − 4 2

Write into the
form of 2 and 2

1 1 = 3
න = න
= 2
3 − 4 2 3 2 − 2 2
= 2
=න 1
∙
2 − 2 2 = 2

11 1 −1 1 2
= 2න = 2 + = 2 −1 3 + #
2 − 2

MOOC MAT438/ UiTM

න 2

3 − 4 6

Write into the
form of 2 and 2

2 2 = 3
න = න
= 2 3
3 − 4 6 3 2 − 2 3 2

2 = 6 2
= න 2 − 2 ∙ 6 2

= 6 2

11 1 −1 1 2 3
= 6න = 6 + = 6 −1 3 + #
2 − 2

MOOC MAT438/ UiTM

MOOC MAT438/ UiTM

2 2
න 4 + 9

Write into the
form of 2 and 2

2 2 = 2න 2 3 2 = 3
න 4 + 9 2 2 +
= 2

= 2 2

2
= 2 න 2 + 2 ∙ 2 2 = 2 2

1 = 1 −1 + = 1 −1 2 + #
= න 2 + 2 3 3

MOOC MAT438/ UiTM


න

4 − 2

Write into the
form of 2 and 2

=න 1 = 2
න
=
4 − 2 2 2 − 2 1
=
1 =
= න ∙

2 − 2

=න 1 = −1 + = −1 + #
2 − 2 2

MOOC MAT438/ UiTM


න

2 − 4

1
න = න

2 − 4 2 − 2 2

1
= න

2 − 2

= 1 −1 + = 1 −1 + #
2 2

MOOC MAT438/ UiTM

MOOC MAT438/ UiTM

3 3 The best way is we choose
න 16 + 23
= 3 because
3 = 3 3 3



Write into the
form of 2 and 2

3 3 3 3 = 4
න 16 + 23 = න 4 2 + 3 2
= 3
3 3
= න 2 + 2 ∙ 3 3 3 = 3 3 3


= 3 3 3

11 = 1 1 −1 + = 1 −1 3 + #
= 3 න 2 + 2 3∙ 12 4

MOOC MAT438/ UiTM

න 1 Inverse Trigonometric or
Inverse Hyperbolic, maybe?

8 − 2 + 1 2

න 1 = න 1 Write into the
8 − 2 + 1 2 2 4 − + 1 2 form of 2 and 2

= 2

11 = + 1
=න
2 2 2 − + 1 2
= 1
11 =
=න
2 2 − 2

= 1 −1 + = 1 −1 + 1 + #
2 2 2

MOOC MAT438/ UiTM

MOOC MAT438/ UiTM

න 1 constant Inverse Trigonometric or
Inverse Hyperbolic, maybe?
14 − 12 − 2 2
Write 14 − 12 − 2 2 into
the form of 2 and 2

By using completing
the square method

14 − 12 − 2 2 = −2 2 − 12 + 14  Re-arrange : 2 + +
= −2 2 + 6 − 7
 if coefficient of 2 not equal to

positive one, then factorize the
coefficient of 2

= −2 2 + 6 + +3 2 − +3 2 − 7  i) Put + − after the term containing x

2 simplify ii) inside bracket must ( × coeffiecient of x)


= −2 + 3 2 − 16 The first three terms → write

The remaining terms → simplify

= 2 16 − + 3 2

MOOC MAT438/ UiTM

න 1

14 − 12 − 2 2

න 1 = න 1 Write into the
form of 2 and 2
14 − 12 − 2 2 2 16 − + 3 2
= 4
11 = + 3
=න
2 16 − + 3 2
= 1
11 =
=න
2 4 2 − + 3 2

11 = 1 −1 + = 1 −1 + 3 + #
=න 2 2 4
2 2 − 2

MOOC MAT438/ UiTM

MOOC MAT438/ UiTM

constant Inverse Trigonometric or
Inverse Hyperbolic, maybe?
න 2 2 − 8 + 26
Write 2 2 − 8 + 26
into the form of 2 and 2 By using completing

the square method

2 2 − 8 + 26 = 2 2 − 4 + 13  Re-arrange : 2 + +

= 2 2 − 4 + −2 2 − −2 2 + 13  if coefficient of 2 not equal to

positive one, then factorize the
coefficient of 2

2 simplify  i) Put + − after the term containing x

= 2 − 2 2 + 9 ii) inside bracket must ( × coeffiecient of x)


The first three terms → write

= 2 − 2 2 + 3 2 The remaining terms → simplify

MOOC MAT438/ UiTM



න 2 2 − 8 + 26

=න 1 Write into the
න 2 2 − 8 + 26 2 form of 2 and 2
− 2 2 + 3 2
= 3
11 = − 2
= 2 න − 2 2 + 3 2

11 = 1
= 2 න 2 + 2 =

= 1 ∙ 1 −1 + = 1 −1 − 2 + #
2 6 3

MOOC MAT438/ UiTM

MOOC MAT438/ UiTM