Question

If \[ \int \frac{1-5\cos^2 x}{\sin^5 x\cos^2 x}\,dx=f(x)+C, \] where \( C \) is the constant of integration, then \[ f\!\left(\frac{\pi}{6}\right)-f\!\left(\frac{\pi}{4}\right) \] is equal to:

• A. \( \dfrac{1}{\sqrt{3}}(26-\sqrt{3}) \)
• B. \( \dfrac{1}{\sqrt{3}}(26+\sqrt{3}) \)
• C. \( \dfrac{4}{\sqrt{3}}(8-\sqrt{6}) \)
• D. \( \dfrac{2}{\sqrt{3}}(4+\sqrt{6}) \)

Hint:

Always try to reduce trigonometric integrals to powers of \( \tan x \) or \( \sec x \); substitutions then become straightforward.

Answer 1

The Correct Option is A