Question

Evaluate \[ \int \frac{dx}{10\cos^2x+\cos2x+5} \]

• A. \(\dfrac14\tan^{-1}\!\left(\dfrac{\tan x}{2}\right)+c\)
• B. \(\dfrac18\tan^{-1}\!\left(\dfrac{\tan x}{2}\right)+c\)
• C. \(\dfrac14\tan^{-1}\!\left(\dfrac{\tan x}{5}\right)+c\)
• D. \(\dfrac18\tan^{-1}\!\left(\dfrac{\tan x}{5}\right)+c\)

Hint:

For integrals containing both \[ \cos^2x \] and \[ \cos2x, \] first use \[ \boxed{\cos2x=2\cos^2x-1,} \] then apply the substitution \[ \boxed{t=\tan x.} \] This converts the integral into the standard form \[ \int\frac{dt}{t^2+a^2}. \]

Answer 1

The Correct Option is B