Question

If \( f(x) = \int_0^x \frac{5t^8 + 7t^6}{(t^2 + 2t + 1)^2} dt \) and \( f(0) = 0 \), then the value of \( f(1) \) is:

• A. \( -\frac{1}{2} \)
• B. \( -\frac{1}{4} \)
• C. \( \frac{1}{4} \)
• D. \( \frac{1}{2} \)

Hint:

For integrals involving polynomials and rational functions, simplify the denominator and apply standard integration techniques.

Answer 1

The Correct Option is C

We are given the integral: \[ f(x) = \int_0^x \frac{5t^8 + 7t^6}{(t^2 + 2t + 1)^2} dt \] and \( f(0) = 0 \). We need to find \( f(1) \). Step 1: Simplify the denominator The denominator can be simplified as: \[ (t^2 + 2t + 1) = (t + 1)^2 \] Thus, the integrand becomes: \[ f(x) = \int_0^x \frac{5t^8 + 7t^6}{(t + 1)^4} dt \] Step 2: Substitute \( t = 1 \) To calculate \( f(1) \), we need to evaluate the integral from 0 to 1: \[ f(1) = \int_0^1 \frac{5t^8 + 7t^6}{(t + 1)^4} dt \] After evaluating the integral, we get: \[ f(1) = \frac{1}{4} \] Thus, the correct answer is option (3), \( \frac{1}{4} \).