Question

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

• A. \( \frac{\pi}{2} \)
• B. \( \frac{\pi}{4} \)
• C. \( \frac{\pi}{3} \)
• D. \( \frac{\pi}{6} \)

Hint:

The integral \( \int \frac{dx}{1 + x^2} \) is a standard result, and it equals \( \tan^{-1}(x) \).

Answer 1

The Correct Option is B

Step 1: Recognizing the integral.
The integral \( \int_0^1 \frac{dx}{1 + x^2} \) is a standard integral. We recognize that: \[ \int \frac{dx}{1 + x^2} = \tan^{-1}(x) \] Step 2: Applying the limits.
Using the limits from 0 to 1, we have: \[ \int_0^1 \frac{dx}{1 + x^2} = \tan^{-1}(1) - \tan^{-1}(0) \] Since \( \tan^{-1}(1) = \frac{\pi}{4} \) and \( \tan^{-1}(0) = 0 \), we get: \[ \frac{\pi}{4} - 0 = \frac{\pi}{4} \] Step 3: Conclusion.
Thus, the value of the integral is \( \frac{\pi}{4} \), corresponding to option (B).