7.3 Integration Techniques
Question #1 (Medium): Evaluating the Integral Using Sine Substitution
When the function is in the form of √ , then use sine substitution where and
. Based on the identity , under the square root simplifies to
. Combined with substitution, apply half angle identity ( )which makes
taking the anti-derivative much easier. Then change the anti-derivative into based
on the double-angle formula so that plugging in the original variable is much easier. Draw the right angle
triangle with angle and the opposite and hypotenuse sides labeled based on the substitution
. Find the adjacent side based on the Pythagorean Theorem, and this expression should
equal to the original expression in the integral √ .
Evaluate the integral.
Since the integral is in the form of √ , sine substitution is appropriate. Given integral can be
written as √ √