Integration By Partial Fractions

We are given:
Step 1: Take LCM on the Left-Hand Side Rewriting the left-hand side with a common denominator: Since the denominators are equal, equating the numerators:
Step 2: Expand the Equation Expanding both terms:
Step 3: Compare Coefficients Since the right-hand side is just , we equate coefficients: 1. For : . 2. For : . 3. For the constant term: .
Step 4: Solve for Substituting into : From the constant term equation: Substituting :
Step 5: Find Thus, the correct answer is:

Step 1: Partial Fraction Decomposition
We are given the integral: We assume a decomposition of the form: Multiplying both sides by to eliminate denominators, we get: Expanding and equating coefficients of like powers of , we determine values of and .
Step 2: Integrating Both Sides

Step 3: Evaluating the Given Expression
Given: By substituting the calculated values of and :
Final Answer: .