7.3 Integration Techniques
Question #4 (Medium): Evaluating Complex Integral Using Trigonometric Substitution
There are three forms of trigonometric substitution as follows:
For √ , use . For√ , use . For √ , use . But if
one of the three is not obvious from the beginning, then check out the expression usually within the
square root (but not always!) and try to convert to perfect square. Then it should fall into one of the
three categories, so then proceed as usual with trig substitution.
Evaluate the integral.
Typical square root is not there. Notice the denominator contains quadratic function. Completing the
square: ( ) ( ) ( ) ( ) √
Then it follows the for√ , so can be used.
Let √ , then √ and √ . Then:∫( )
∫ (√ ) ∫ (√ ) √ ∫ √ √ . Based on
(( ) √ ) ((√ ) √ )