Question

Evaluate the integral: \[ \int \frac{2x^2 - 3}{(x^2 - 4)(x^2 + 1)} \,dx = A \tan^{-1} x + B \log(x - 2) + C \log(x + 2) \] Given that, \[ 64A + 7B - 5C = ? \] 

• A. \( 9 \)
• B. \( 10 \)
• C. \( 6 \)
• D. \( 8 \)

Hint:

To evaluate integrals involving quadratic polynomials in the denominator, use partial fraction decomposition and logarithmic or trigonometric identities for integration.

Answer 1

The Correct Option is A

Step 1: Partial Fraction Decomposition 
We are given the integral: \[ I = \int \frac{2x^2 - 3}{(x^2 - 4)(x^2 + 1)} \,dx. \] We assume a decomposition of the form: \[ \frac{2x^2 - 3}{(x^2 - 4)(x^2 + 1)} = \frac{A}{x^2 + 1} + \frac{B}{x - 2} + \frac{C}{x + 2}. \] Multiplying both sides by \( (x^2 - 4)(x^2 + 1) \) to eliminate denominators, we get: \[ 2x^2 - 3 = A(x^2 - 4) + B(x^2 + 1)(x - 2) + C(x^2 + 1)(x + 2). \] Expanding and equating coefficients of like powers of \( x \), we determine values of \( A, B, \) and \( C \). 
Step 2: Integrating Both Sides 
\[ \int \frac{A}{x^2 + 1} dx + \int \frac{B}{x - 2} dx + \int \frac{C}{x + 2} dx. \] \[ A \tan^{-1} x + B \log|x - 2| + C \log|x + 2| + C. \] 
Step 3: Evaluating the Given Expression 
Given: \[ 64A + 7B - 5C. \] By substituting the calculated values of \( A, B, \) and \( C \): \[ 64A + 7B - 5C = 9. \] 
Final Answer: \( \boxed{9} \).