Question

Evaluate the integral \[ \int \frac{x^5}{x^2 + 1} dx. \]

• A. \( \frac{x^4}{4} + \frac{x^3}{3} - \tan^{-1} x + c \)
• B. \( \frac{x^4}{4} - \frac{x^2}{2} + \frac{1}{2} \log(x^2 + 1) + c \)
• C. \( \frac{x^4}{4} + \frac{x^3}{3} + \tan^{-1} x + c \)
• D. \( \frac{x^4}{4} + \frac{x^2}{2} - \frac{1}{2} \log(x^2 + 1) + c \)

Hint:

For rational integrals, split terms and use substitution if needed.

Answer 1

The Correct Option is B

Step 1: Splitting the integral Rewriting: \[ I = \int \frac{x^5}{x^2 + 1} dx. \] Splitting: \[ I = \int x^3 \cdot \frac{x^2}{x^2 + 1} dx. \] Step 2: Using substitution Let \( u = x^2 + 1 \), then \( du = 2x dx \). This simplifies the integral, leading to: \[ I = \frac{x^4}{4} - \frac{x^2}{2} + \frac{1}{2} \log(x^2 + 1) + c. \]