# Integral of $\int\frac{\sin x}{\sin x+\cos x}dx$

The questions defines $$I=\int\frac{\sin x}{\sin x +\cos x}dx\;\;J=\int\frac{\cos x}{\sin x +\cos x}dx$$ It asked me to find $$I+J$$ and $$J-I$$ which I have done and I will show below but now I need to find the integral shown below and I'm unsure on what to do. $$\int\frac{\sin x}{\sin x+\cos x}dx$$

I have found that: $$I+J = x+c$$ $$J-I=\ln{|\cos x +\sin x|} +c$$

But now i'm unsure on how to find just $$I$$

• If you know what $J-I$ is, you know what $I-J$ is. Try adding $I-J$ and $I+J$ together. – welshman500 Dec 12 '18 at 17:36

what you have is: $$I+J=x+C$$ $$I-J=-\ln|\cos(x)+\sin(x)|+C$$ so: $$2I=x-\ln|\cos(x)+\sin(x)|+C$$
$$\begin{cases} J+I = x+c \\ J - I = \ln|\cos x + \sin x | + c\end{cases}$$
See an easy way out for $$I$$ ?