Question

\[ \int \frac{x^2}{(x \sin x + \cos x)^2} dx \text{ is equal to} \]

• A. \(\frac{x \sin x - \cos x}{x \sin x + \cos x} + C\)
• B. \(\frac{\cos x - \sin x}{x \sin x + \cos x} + C\)
• C. \(\frac{x \cos x - \sin x}{x \sin x + \cos x} + C\)
• D. \(\frac{\sin x - x \cos x}{x \sin x + \cos x} + C\)

Hint:

Thermal neutrons have kinetic energies around 0.025 eV at room temperature.

Answer 1

The Correct Option is D

Let \(I = \int \frac{x^2}{(x \sin x + \cos x)^2} dx\). Let the denominator be \(f(x) = x \sin x + \cos x\). Then: \[ f'(x) = \sin x + x \cos x - \sin x = x \cos x \] Let numerator = derivative of numerator \(g(x) = \sin x - x \cos x\) Then: \[ g'(x) = \cos x + x \sin x - \cos x = x \sin x \] Try differentiating option (d), you will get the integrand. Hence verified.