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Lecture

# 7.3 Trigonometric Substitution Question #1 (Easy)

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University of Toronto St. George

Mathematics

MAT136H1

all

Winter

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7.3 Integration Techniques
Trigonometric Substitution
Question #1 (Medium): Evaluating the Integral Using Sine Substitution
Strategy
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 √ .
Sample Question
Evaluate the integral.
∫√
Solution
Since the integral is in the form of √ , sine substitution is appropriate. Given integral can be
written as √ √

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