Question

Evaluate \[ \int_0^{\frac{\pi}{2}} \frac{\sin x}{1 + \cos^2 x} \, dx \] is

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

Hint:

For integrals involving trigonometric functions, use substitution to simplify and find the antiderivative.

Answer 1

The Correct Option is B

Step 1: Apply trigonometric substitution.
Use a substitution to simplify the integrand. The integral \( \int_0^{\frac{\pi}{2}} \frac{\sin x}{1 + \cos^2 x} \, dx \) can be solved by using standard integration techniques.
Step 2: Conclusion.
After performing the integration, the result is \( \frac{\pi}{4} \). Final Answer: \[ \boxed{\frac{\pi}{4}} \]