Integrals Related to Inverse Trig,
Inverse Hyperbolic Functions
Lesson 9.5
Review
• Recall derivatives of inverse trig functions
d
1 du
1
sin u 
, u 1
dx
1  u 2 dx
d
1 du
1
tan u 
dx
1  u 2 dx
d
1
du
1
sec u 
, u 1
2
dx
u u  1 dx
2
Integrals Using Same
Relationships
du
u
 a 2  u 2  arcsin a  C
du
1
u
 a 2  u 2  a arctan a  C
du
1
u
 u u 2  a 2  a arcsec a  C
When given
integral problems,
look for these
patterns
3
Identifying Patterns
• For each of the integrals below, which
inverse trig function is involved?
4dx
 13  16 x 2

dx
9 x
2
x
dx
25 x 2  4
dx
 x 2  2 x  10
Hint: use
completing
the square
4
Warning
• Many integrals look like the inverse trig
forms
• Which of the following are of the inverse
trig forms?

x dx
 1  x2
dx
 1  x2

x dx
1  x2
dx
1 x
2
If they are not,
how are they
integrated?
5
Try These
• Look for the pattern or how the expression
can be manipulated into one of the
patterns

8dx
 1  16 x 2

x dx
1  25 x 2
dx
4 x 2  4 x  15

x 5
x  10 x  16
2
dx
6
Integrals Involving Inverse
Hyperbolic Functions


1
u
du  sinh
C
a
u 2  a2
1
1 u
du  cosh
C
a
u2  a2
1
1
1
-1 u
 a 2  u 2 du  a tanh a  C
1
1
1 u
 u a 2  u 2 du  a sech a  C
7
Try It!
4
• Note the definite integral

1
1
9  4 x2
dx
• What is the a, the u, the du?

a = 3, u = 2x, du = 2 dx
4
1
2
dx

2 1 9  4 x 2
4
1
1 2 x
 sinh
2
3 1
8
Application
• Find the area enclosed by
x = -¼, x = ¼, y = 0, and
y
1
1  4x2
• Which pattern does this match?
• What is the a, the u, the du?
9
Assignment
• Lesson 9.5
• Page 381
• Exercises 1 – 35 odd
10