Question

Evaluate the integral \( I = \int_0^\pi \frac{4 \cos^2 x + \sin^2 x}{8x} \, dx \).

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

Hint:

To evaluate integrals with trigonometric functions, use trigonometric identities to simplify the expressions before integration.

Answer 1

The Correct Option is C

We solve the integral \( I \) as follows: \[ I = \int_0^\pi \frac{4 \cos^2 x + \sin^2 x}{8x} \, dx \] First, we split the integral into two parts: \[ I = \int_0^\pi \frac{4 \cos^2 x}{8x} \, dx + \int_0^\pi \frac{\sin^2 x}{8x} \, dx \] Simplify and evaluate each part. After calculating, the final value of the integral is \( 2\pi^2 \).