Question

Evaluate ; ${\int }_0^a\frac{\sqrt{x}}{\sqrt{x}+\sqrt{a-x}}dx.$

Answer 1

${\int }_0^a\frac{\sqrt{x}}{\sqrt{x}+\sqrt{a-x}}dx$

$f\left(x\right)={\int }_0^a\frac{\sqrt{x}}{\sqrt{x}+\sqrt{a-x}}dx$

$f(a-x)={\int }_0^a\frac{\sqrt{a-x}}{\sqrt{a-x}+\sqrt{x}}dx$

$\Rightarrow f\left(x\right)=f(a-x)[A{\int }_0^{a}f\left(x\right)=f(a-x)[A{\int }_0^{a}f(a-x)]$

$\Rightarrow 2f\left(x\right)=1$

$f\left(x\right)=1/2$