Question

$\int \frac{\sin x}{\cos x(1+\cos x)}dx=f\left(x\right)+c\Rightarrow f\left(x\right)=$

• A. $\log |\frac{1+\cos x}{\cos x}|$
• B. $\log |\frac{\cos x}{1+\cos x}|$
• C. $\log |\frac{\sin x}{1+\sin x}|$
• D. $\log |\frac{1+\sin x}{\sin x}|$

Answer 1

The correct option is A $\log |\frac{1+\cos x}{\cos x}|$

$\int \frac{\sin xdx}{\cos x(1+\cos x)}dx$
$=-\int \frac{d\cos x}{\cos x(1+\cos x)}$
$=\int \frac{\left(\cos x\right)-(1+\cos x)}{\left(\cos x\right)(1+\cos x)}d\cos x$
$=\int \frac{d\cos x}{1+\cos x}-\int \frac{d\cos x}{\cos x}$
$=\log (1+\cos x)-\log \left(\cos x\right)+k$
$=\log \mid \frac{1+\cos x}{\cos x}\mid +c$