I’m having trouble computing the integral:

$$\int \frac{\sin(x)}{\sin(x)+\cos(x)}\mathrm dx.$$

I hope that it can be expressed in terms of elementary functions. I’ve tried simple substitutions such as $u=\sin(x)$ and $u=\cos(x)$, but it was not very effective.Any suggestions are welcome. Thanks.

**Answer**

Let $I:=\int\frac{\cos x}{\cos x+\sin x}dx$ and $J:=\int\frac{\sin x}{\cos x+\sin x}dx$. Then $I+J=x + C$, and

$$I-J=\int\frac{\cos x-\sin x}{\cos x+\sin x}dx=\int\frac{u'(x)}{u(x)}dx,$$

where $u(x)=\cos x+\sin x$. Now we can conclude.

**Attribution***Source : Link , Question Author : Michael Li , Answer Author : Elliptic Curve*