Question

Solve:

${\int }_0^{\frac{\pi }{2}}\frac{\sin x}{\sin x+\cos x}dx$

Answer 1

We have,

$I={\int }_0^{\frac{\pi }{2}}\frac{\sin x}{\sin x+\cos x}dx$ $........\left(1\right)$

We know that

${\int }_a^{b}f\left(x\right)dx$$={\int }_a^{b}f(a+b-x)dx$

Therefore,

$I={\int }_0^{\frac{\pi }{2}}\frac{\cos x}{\cos x+\sin x}dx$ $........\left(2\right)$

On adding equations $\left(1\right)$ and $\left(2\right)$, we get

$2I={\int }_0^{\frac{\pi }{2}}\frac{\sin x+\cos x}{\sin x+\cos x}dx$

$2I={\int }_0^{\frac{\pi }{2}}1dx$

$2I={\left[x\right]}_0^{\frac{\pi }{2}}$

$2I=\frac{\pi }{2}$

$I=\frac{\pi }{4}$

Hence, this is the answer.