Question

If \( f(x) = \int_{x^2}^{\cos^2 x} (2x \tan^2 x - 2x - 6 \tan x) dx \), and \( f(0) = \pi \), then \( f(x) = \):

• A. \( x^2 \sin x + \pi \)
• B. \( \cos x + \pi - 1 \)
• C. \( -x^3 \sin 2x + \pi \)
• D. \( x^3 \cos 2x + \cos x \)

Hint:

Use Leibniz’s Rule when the integral has variable limits and a known boundary condition.

Answer 1

The Correct Option is C

Apply Leibniz rule for differentiation under the integral with variable limits. Let \[ f(x) = \int_{x^2}^{\cos^2 x} g(x) dx \] Then \[ f(x) = -\int_{\cos^2 x}^{x^2} g(x) dx = -x^3 \sin 2x + \pi \] using integral simplification and given boundary value.