Question

Solve:
$\int \frac{sinx}{(\sin x-\cos x)}dx$

Answer 1

$\begin{array}{c}wehave\\ I=\int \frac{\sin x}{\sin x-\cos x}dx\\ multiplingthenumeratoranddenominatorby\sin x+cosxweget\\ =\int \frac{\sin x\left(\sin x+\cos x\right)}{\left(\sin x-\cos x\right)\left(\sin x+\cos x\right)}dx\\ =\int \frac{{\sin }^{2}x+\sin x\cos x}{{\sin }^{2}x-{\cos }^{2}x}dx\\ =\int \frac{{\sin }^{2}x}{{\sin }^{2}x-{\cos }^{2}x}dx+\int \frac{\sin x\cos x}{{\sin }^{2}x-{\cos }^{2}x}dx\\ u\sin g{\cos }^{2}x-{\sin }^{2}x=\cos 2x,\sin 2x=2\sin x\cos xand{\sin }^{2}x=\frac{1-\cos 2x}{2}weget\\ =-\int \frac{1-\cos 2x}{2\cos 2x}dx-\int \frac{\sin 2x}{2\cos 2x}dx\\ =-\int \frac{\sec 2x}{2}dx+\int \frac{1}{2}dx-\int \frac{\tan 2x}{2}dx\\ u\sin g\int \tan xdx=\log |\sec x|+c.,\int \sec xdx=\log |\sec x+\tan x|+C\\ and,\int dx=x+C\\ weget\\ \frac{\log |\sec 2x+\tan 2x|}{4}+\frac{x}{2}-\frac{\log s|ec2x|}{4}+C.\end{array}$