7.3 Integration Techniques
Question #2 (Medium): Evaluating the Integral Using Tangent Substitution
For the expressio√ let . Then . Substitute into the expression
√ ( ) √ √ √ based on the identity
. Then combined with other expressions, simplify the integral and then evaluate
using various tricks learned so far.
Evaluate the integral.
First, denominator needs to be expressed in terms of squares. Perfecting the squares: √
√ ( ) √ ( ) , then tangent substitution can be used, since it follows after the form
√ . Then, ; , and . So substitute to the
integral∫ ∫ ∫ ∫
√ √( ) √ ( ) √
∫ . Based on