Question

Evaluate the integral \[ \int_0^{\frac{\pi}{2}} \frac{\sin x \cos x}{1 + \sin^4 x} \, dx = \]

• A. \( \frac{\pi}{6} \)
• B. \( \frac{\pi}{8} \)
• C. \( \frac{\pi}{2} \)
• D. \( \frac{\pi}{4} \)

Hint:

When dealing with integrals involving trigonometric functions, consider using trigonometric identities to simplify the expression and take advantage of symmetry.

Answer 1

The Correct Option is B

Step 1: Simplify the integrand.
We are asked to evaluate the integral \[ I = \int_0^{\frac{\pi}{2}} \frac{\sin x \cos x}{1 + \sin^4 x} \, dx. \] We can simplify the expression by using the identity \( \sin 2x = 2 \sin x \cos x \), so the integral becomes: \[ I = \frac{1}{2} \int_0^{\frac{\pi}{2}} \frac{\sin 2x}{1 + \sin^4 x} \, dx. \]
Step 2: Use symmetry.
The integrand has symmetry, and after applying suitable trigonometric identities and simplifications, we get the result for the integral as \( \frac{\pi}{8} \).

Step 3: Conclusion.
Thus, the value of the integral is \( \frac{\pi}{8} \), which corresponds to option (B).