Question

Evaluate the integral \[ \int_{0}^{\pi} x \cdot \sin x \cdot \int_{x}^{5} \frac{\cos x}{x} \cdot dx = \]

• A. \( \dfrac{16\pi}{693} \)
• B. \( \dfrac{8\pi}{693} \)
• C. \( \dfrac{4\pi}{693} \)
• D. \( \dfrac{2\pi}{693} \)

Hint:

When dealing with nested integrals, be careful to identify which variable is the one being integrated at each step and apply the limits accordingly.

Answer 1

The Correct Option is B

This is a nested definite integral. The inner integral \( \int_{x}^{5} \frac{\cos x}{x} \, dx \) is treated as a constant with respect to the outer integral variable \(x\). Using properties of definite integrals and solving step-by-step through substitution or integration by parts, we arrive at the result \( \dfrac{8\pi}{693} \).