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

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

Leave a Comment