Question

integration with limits (0 to a) root(x)/ root(x) + root(a-x)

Answer 1

$$\begin{gathered} \mathrm{Let}I={\int }_o^a\frac{\sqrt{x}}{\sqrt{x}+\sqrt{a-x}}dx.....\left(1\right) \\ \Rightarrow I={\int }_o^a\frac{\sqrt{\left(a-x\right)}}{\sqrt{\left(a-x\right)}+\sqrt{a-\left(a-x\right)}}dx \\ \Rightarrow I={\int }_o^a\frac{\sqrt{a-x}}{\sqrt{a-x}+\sqrt{a-a+x}}dx \\ \Rightarrow I={\int }_o^a\frac{\sqrt{a-x}}{\sqrt{a-x}+\sqrt{x}}dx \\ \Rightarrow I={\int }_o^a\frac{\sqrt{a-x}}{\sqrt{x}+\sqrt{a-x}}dx.....\left(2\right) \\ \mathrm{Adding}\left(1\right)\mathrm{and}\left(2\right)\mathrm{we}\mathrm{get} \\ 2I={\int }_o^a\frac{\sqrt{x}+\sqrt{a-x}}{\sqrt{x}+\sqrt{a-x}}dx \\ \Rightarrow 2I={\int }_o^a\left(1\right)dx \\ \Rightarrow 2I={\left[x\right]}_0^a \\ \Rightarrow 2I=\left(a-0\right) \\ \Rightarrow 2I=a \\ \therefore I=\frac{a}{2} \end{gathered}$$