Question

Evaluate \[ \int \frac{\sin^2x\tan x} {\cos^6x+\sin^4x\cos^2x+\cos^4x\sin^2x+\sin^6x}\,dx. \]

• A. \[ \log(\sin^4x+\cos^4x)+c \]
• B. \[ \frac14\log(\sin^4x+\cos^4x)+c \]
• C. \[ \frac14\log(1+\tan^4x)+c \]
• D. \[ \log(1+\tan^4x)+c \]

Hint:

Whenever the denominator contains \[ 1+\tan^2x+\tan^4x+\tan^6x, \] use the factorization \[ \boxed{ 1+\tan^2x+\tan^4x+\tan^6x = (1+\tan^2x)(1+\tan^4x), } \] followed by the substitution \[ \boxed{t=\tan x.} \]

Answer 1

The Correct Option is C