Question

Evaluate the integral \( \int_0^1 \frac{x^2}{1 + x^2} dx \)

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

Hint:

For integrals involving rational functions, consider trigonometric substitution to simplify the expression.

Answer 1

The Correct Option is B

Step 1: Apply integration techniques.
We use the substitution \( x = \tan(\theta) \), so \( dx = \sec^2(\theta) d\theta \). The limits change accordingly, and we integrate the resulting expression: \[ \int_0^1 \frac{x^2}{1 + x^2} dx = \int_0^{\frac{\pi}{4}} \tan^2(\theta) d\theta \] Step 2: Simplify the result.
After evaluating the integral, we find: \[ \int_0^1 \frac{x^2}{1 + x^2} dx = 1 - \frac{\pi}{4} \] Step 3: Conclusion.
The correct answer is (B) \( 1 - \frac{\pi}{4} \).