Inverse Trigonometric Functions:
Integration
Lesson 5.8
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

x
dx
25 x 2  4
dx
9 x
2
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
6
dx
Completing the Square
• Often a good strategy when quadratic
functions are involved in the integration
dx
 x 2  2 x  10
• Remember … we seek
(x – b)2 + c
 Which might give us an integral resulting in the
arctan function
Completing the Square
• Try these
2
dx
2 x2  4 x  13

2
x  4x
2
dx
Rewriting as Sum of Two Quotients
• The integral may not appear to fit basic
integration formulas
 May be possible to split the integrand into two
portions, each more easily handled

4x  3
1 x
2
dx
Basic Integration Rules
• Note table of basic rules
 Page 364
• Most of these should be committed to
memory
• Note that to apply these, you must create
the proper du to correspond to the u in the
formula
 cos u du  sin u  C
Assignment
• Lesson 5.8
• Page 366
• Exercises 1 – 39 odd
63, 67
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