Question

$\int \frac{\cos x}{{\cos }^{2}x+5{\sin }^{2}x}dx$ is equal to

• A. $\log (\sqrt{\left(2\sin x\right)}+C$
• B. ${\tan }^{-1}\left(\sin x\right)+C$
• C. $\frac{1}{2}{\tan }^{-1}\left(2\sin x\right)+C$
• D. $\frac{1}{4}{\tan }^{-1}\left(2\sin 2x\right)+C$

Answer 1

The correct option is C $\frac{1}{2}{\tan }^{-1}\left(2\sin x\right)+C$

$\int \frac{\cos x}{{\cos }^{2}x+{\sin }^{2}x+4{\sin }^{2}x}dx=\int \frac{\cos x}{1+4{\sin }^{2}x}dx$
Let $t=\sin x\Rightarrow dt=\cos xdx$
$=\frac{1}{4}\int \frac{1}{{\left(\frac{1}{2}\right)}^2+t^2}dt=\frac{2}{4}{\tan }^{-1}2t+C=\frac{1}{2}{\tan }^{-1}\left(2\sin x\right)+C$