Question

Evaluate the integral of
$\frac{\cos x}{\left(1-\sin x\right)\left(2-\sin x\right)}$

• A. $\ln (1-\sin x)+\ln (-\sin x+2)+C$
• B. $-\ln (1-\sin x)+\ln (-\sin x+2)+C$
• C. $-\ln (1-\cos x)+\ln (-\cos x+2)+C$
• D. None

Answer 1

The correct option is B $-\ln (1-\sin x)+\ln (-\sin x+2)+C$

$f\frac{\cos x}{(1-\sin x)(2-\sin x)}dx$

$(2+\sin x)=4$

$du=\cos xdx$

$f\frac{du}{u(u-1)}$

$=f[\frac{-1}{u}+\frac{1}{u-1}]du$

$=f\frac{-1}{u}du+f\frac{1}{u-1}du$

$=-\log \left(u\right)+log(u-1)+c$

$=-\log (2+\sin x)+\log (1+\sin x)+c$

$=\log (1-\sin x)-\log (2+\sin x)+c$