Not able to integrate $ \int \frac{\sin x}{\sin x + \cos x} \ dx $ ??

Question

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

So what I thought of doing was converting $\sin x$ and $\cos x$ into $\tan\frac{x}{2}$

But it got converted into non integrable form

Any other methods would be appreciated .

Answer 1

Hint:

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

$J+I$ is pretty easy. For $J-I$ put $\sin x+\cos x= u$ to get numerator of $J-I$ as $du$