Question

Let $f(x)=\tan x+2\tan 2x+4\tan 4x+8\cot 8x$, then primitive of $f(x)$ with respect to $x$ is
where $C$ is constant of integration.

• A. $8\ln (\sin 8x)+C$
• B. $\ln (\sin 8x)+C$
• C. $\ln (\sin x)+C$
• D. $\ln (\sec x)+C$

Answer 1

Answer: (C)

$\int f(x)dx=\ln (\sec x)+\ln (\sec 2x)+\ln (\sec 4x)+\ln (\sin 8x)+C$
$=\ln \left(\frac{\sin 8x}{\cos x\cos 2x\cos 4x}\right)+C=\ln \left(\frac{8\sin x\cdot \sin 8x}{8\sin x\cdot \cos x\cos 2x\cos 4x}\right)$
$=\ln \left(\frac{8\sin x\sin 8x}{\sin 8x}\right)=\ln (8\sin x)=\ln 8+\ln (\sin x)+C=\ln (\sin x)+C$