Integration By Parts

Step 1: Express in terms of sine and cosine. We rewrite the given expression using sine and cosine functions: Thus, the integral becomes: Step 2: Rewrite in a simplified form. We factor and simplify: Using the identity: we rewrite: Step 3: Use trigonometric substitution. Using the identity: we get: Thus, the integral simplifies to: Step 4: Substituting . Let: Rewriting in terms of , we simplify and integrate: Using integration techniques, solving for , and substituting back gives the final result.

To evaluate the integral: We use the method of partial fractions. We begin by expressing the integrand as: Multiplying both sides by to clear the denominators: Expanding both sides: Now, equate the coefficients of and on both sides. For the -terms: For the -terms: Solving this system of equations: From , we have . Substituting this into the second equation: Since , we have: Thus, the partial fractions decomposition is: Now, integrate each term: Thus, the integral is: