Download Trigonometric Substitutions

TRIGONOMETRIC SUBSTITUTION
Find : cos , sin  , tan
3
9  x2

x
TRIGONOMETRIC SUBSTITUTION
4 x
x  2 sin 
Let :
2
4 x 
2
4  (2 sin  )
2
 4  4 sin 
2

4(1  sin  )
NOTE:
We convert the radical
into a cos function
2
 4 cos 2 
 2 cos 
 2 cos
if :
π

2
θ 
π
2
TRIGONOMETRIC SUBSTITUTION
x  25
Let :
x  5 tan 
x 9
Let :
x  3sec
2
2
TRIGONOMETRIC SUBSTITUTION
Example
Example
Find
Example



4  x 2 dx
Find
2
x
4

x
dx

Find
9  x2
dx
2
x
1
dx
x 2  16
x

1
2
4  x2
1
x 2  16
dx
dx
The main idea is how to get rid of the radicals.
( use trig. Substitution)
TRIGONOMETRIC SUBSTITUTION
Expression
a2  x2
a x
2
2
x2  a2
x  a sin 
x  a sin1 sin 2   cos 2 
x  a tan 
x  a tan  x  a tan 
x  a sec 
x  a sec  x  a sec 
TRIGONOMETRIC SUBSTITUTION
Example
Find

1
9 x
2
dx
In the following table we list trigonometric substitutions that are effective for the
given radical expressions because of the specified trigonometric identities.
TRIGONOMETRIC SUBSTITUTION
Example
Find
3
2
x
9

x
dx

In the following table we list trigonometric substitutions that are effective for the
given radical expressions because of the specified trigonometric identities.
TRIGONOMETRIC SUBSTITUTION
Example
Find

9  x2
dx
2
x
TRIGONOMETRIC SUBSTITUTION
Example
Find
x 3dx
 (4 x 2  9)3/ 2
x  32 tan 
TRIGONOMETRIC SUBSTITUTION
EXAM-2
Term-092
TRIGONOMETRIC SUBSTITUTION
Example
Find

x
x 4
2
dx
TRIGONOMETRIC SUBSTITUTION
EXAM-2
Term-102
TRIGONOMETRIC SUBSTITUTION
Example
Find

dx
x2  a2
a0
x  a cosh t
hyperbolic substitutions can be used in place of trigonometric substitutions
and sometimes they lead to simpler answers. But we usually use
trigonometric substitutions because trigonometric identities are more familiar
than hyperbolic identities.
TRIGONOMETRIC SUBSTITUTION
Example
Find

x
3  2x  x
2
dx