Trigonometric Substitutions

Trig. substitutions are usually used for integrals that involve square roots of quadratic expressions; the trig substitution is used to eliminate the radical sign; the following substitutions are made:

  Term involve in the Integral	  Term involve in the Integral	Trig. Identity
Â	 u = a sin q	 cos2 q = 1 - sin2 q
Â	 u = a tan q	 sec2 q = 1 + tan2 q
Â	 u = a sec q	 tan2 q = sec q -1

where a is any real number and u is any variable.

ex.

Substitute: x = sec q

dx = sec q tan q dq

 

Substitute:
u = sin q
du = cos q dq

The solution must be in terms of x:
From the substitution of x = sec q, the diagram of a right triangle gives;

The final solution in terms of x is: