Math 140, Calculus 2
Trigonometric Substitution
n
2
For integrals of the form (a  x ) we can often use trigonometric functions to facilitate
integration. First let us remember some of the trigonometric identities:
2
2
sin 2 x  cos 2 x  1
tan 2 x  1  sec 2 x
For our first example, let us look at the following integral:
dx

a2  x2
What we’ll do here is make “a” the hypotenuse of a right triangle and x one of the sides to give
us this function in terms of trigonometric functions.
a
x
θ
a2  x2
So we have the following relations:
cos  
a2  x2
,
a
x
,
a
x  a sin 
dx  a cos( )d
sin  
If we substitute these values into our original integral we get,

dx
a2  x2

a cos( )d
  d    C
a cos( )
Into theta we insert the inverse sine function and get:
dx
x
 a 2  x 2  arcsin( a )  C .