Question

Evaluate :

$\underset{0}{\overset{\pi /2}{\int }}\frac{\cos xdx}{\left(\cos x+\sin x\right)}$

Answer 1

We have,

$I=\underset{0}{\overset{\pi /2}{\int }}\frac{\cos xdx}{\left(\cos x+\sin x\right)}$ $.........\left(1\right)$

We know that

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

Therefore,

$I=\underset{0}{\overset{\pi /2}{\int }}\frac{\sin xdx}{\left(\sin x+\cos x\right)}$ $............\left(2\right)$

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

$2I=\underset{0}{\overset{\pi /2}{\int }}\frac{\sin x+\cos x}{\left(\sin x+\cos x\right)}dx$

$2I=\underset{0}{\overset{\pi /2}{\int }}1dx$

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

$2I=\frac{\pi }{2}-0$

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

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

Hence, this is the answer.