Question

Integrate: $\int \frac{\sin x}{\sin x+\cos x}dx$

Answer 1

Consider the given integral.

$I=\int \frac{\sin x}{\sin x+\cos x}dx$

$I=\frac{1}{2}\int \frac{2\sin x}{\sin x+\cos x}dx$

$I=\int \frac{1}{2}-\frac{\cos x-\sin x}{2(\sin x+\cos x)}dx$

$I=\int \frac{1}{2}dx-\int \frac{\cos x-\sin x}{2(\sin x+\cos x)}dx$

Put $t=\sin x+\cos x$

$dt=\cos x-\sin xdx$

Therefore,

$I=\int \frac{1}{2}dx-\frac{1}{2}\int \frac{dt}{t}$

$I=\frac{1}{2}x-\frac{1}{2}\ln \left(t\right)+C$

Putting the value of $t$ in the above expression, we get

$I=\frac{1}{2}x-\frac{1}{2}\ln (\sin x+\cos x)+C$