Question

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

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

Answer 1

Answer: (C)

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