2.7 integration of inverse hyperbolic functions

න 1 = ℎ−1 +
2 + 2

න 1 = ℎ−1 +
2 − 2

 න 2 1 2 = 1 ℎ−1 +
−

න 2 1 = 1 ℎ−1 +
− 2
1 + =
= 2 ln − + =

MOOC MAT438/ UiTM

න 1

4 + 9 2 Inverse Trigonometric
or Inverse Hyperbolic,

maybe?

Write into

the form of
2 and 2

න 1 1
=න = 2
4 + 9 2 = 3
2 2 + 3 2

=න 1
∙ = 3
2 + 2 3

= 3

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

2 + 2

MOOC MAT438/ UiTM

MOOC MAT438/ UiTM

න 3 write the denominator in the form

2 − 3 of (which is also found in the

numerator) because =


3 Write into = 3
න 2 − 3 = 3 න 2 − 32 the form of
2 and 2

=

=
= 3 න 2 − 2 ∙

=

= 3න 1 = 3 ℎ−1 + = 3 ℎ−1 + #
3

2 − 2

MOOC MAT438/ UiTM

(in denominator) is actually 1


2 (which the derivative of ).
න 4 − 2
So the best way is we choose

= because = 1


Write into the
form of 2 and 2

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

1 1 + 1 + 2
= 2 න 2 − 2 = 2 ∙ 2 − + = 2 − 2 + #

MOOC MAT438/ UiTM

MOOC MAT438/ UiTM

Inverse Trigonometric or
Inverse Hyperbolic, maybe?

constant Write 4 2 − 24 + 11
into the form of 2 and 2
න
 if coefficient of 2 not equal
4 2 − 24 + 11 to positive one, then factorize

 Re-arrange : 2 + + the coefficient of 2 By using completing
the square method

4 2 − 24 + 11 =4 2 − 6 + 11
4

=4 2 − 6 + −3 2− −3 2 + 11
4
 i) Put + − after the term containing x

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


=4 −3 2 25
−4
The first three terms → write

52 The remaining terms → simplify
2
=4 − 3 2 −

MOOC MAT438/ UiTM

Write into the
form of 2 and 2
න

4 2 − 24 + 11

න = න 1
4 2 − 24 + 11
4 5 2
2
− 3 2 − = 5
2

1 1 = − 3
= 2න

5 2 = 1
2 =
− 3 2 −

1 1 = 1 ℎ−1 − 3 + = 1 ℎ−1 2 − 3 + #
= 2න 2 − 2 2 5 2 5

2

MOOC MAT438/ UiTM

MOOC MAT438/ UiTM

Test way is we choose

ℎ5 ℎ5 = ℎ5 because
න 25 − ℎ25
ℎ5 = −5 ℎ5 ℎ5


Write into the
form of 2 and 2

ℎ5 ℎ5 ℎ5 ℎ5 = 5
න 25 − ℎ25 = න 5 2 − ℎ5 2
= ℎ5
ℎ5 ℎ5
= න 2 − 2 ∙ −5 ℎ5 ℎ5 = −5 ℎ5 ℎ5


= −5 ℎ5 ℎ5

= 1 1 1 ∙ 1 ℎ−1 + = − 1 ℎ−1 ℎ5 + #
− 5 න 2 − 2 = −5 25 5

MOOC MAT438/ UiTM

න 1 Inverse Trigonometric or
Inverse Hyperbolic, maybe?
4 + 2 − 1 2

න 1 = න 1 Write into the
form of 2 and 2

4 + 2 − 1 2 2 2 + − 1 2

11 = 2
=න = − 1
2 2 + − 1 2

1 1 = 1
=න =

2 2 2 + − 1 2

= 1 1 = 1 ℎ−1 + = 1 ℎ−1 − 1 + #
න 2
22
2 2 + 2

MOOC MAT438/ UiTM

MOOC MAT438/ UiTM