Question

Integrate it using properties of definite integrals
PLS DO THIS, PLS

Answer 1

Answer:

$\int\limits^a_0{\frac{\sqrt{x}}{\sqrt{x}+\sqrt{a-x}}}\,dx=\frac{a}{2}$

Step-by-step explanation:

I think your question may be

$\int\limits^a_0{\frac{\sqrt{x}}{\sqrt{x}+\sqrt{a-x}}}\,dx$

$\text{Let }I=\int\limits^a_0{\frac{\sqrt{x}}{\sqrt{x}+\sqrt{a-x}}}\,dx$.....(1)

$\text{Using the property}$

$\boxed{\bf\int\limits^a_0{f(x)}\,dx=\int\limits^a_0{f(a-x)}\,dx}$

$I=\int\limits^a_0{\frac{\sqrt{a-x}}{\sqrt{a-x}+\sqrt{a-(a-x)}}}\,dx$

$I=\int\limits^a_0{\frac{\sqrt{a-x}}{\sqrt{a-x}+\sqrt{x}}}\,dx$

$I=\int\limits^a_0{\frac{\sqrt{a-x}}{\sqrt{x}+\sqrt{a-x}}}\,dx$........(2)

$\text{Adding (1) and (2)}$

$2I=\int\limits^a_0{\frac{\sqrt{x}}{\sqrt{x}+\sqrt{a-x}}}\,dx+\int\limits^a_0{\frac{\sqrt{a-x}}{\sqrt{x}+\sqrt{a-x}}}\,dx$

$\text{combining the integrals, we get}$

$2I=\int\limits^a_0[\frac{\sqrt{x}}{\sqrt{x}+\sqrt{a-x}}+\frac{\sqrt{a-x}}{\sqrt{x}+\sqrt{a-x}}]dx$

$2I=\int\limits^a_0\frac{\sqrt{x}+\sqrt{a-x}}{\sqrt{x}+\sqrt{a-x}}\,dx$

$2I=\int\limits^a_0\,dx$

$2I=[x]^a_0$

$2I=a$

$\implies\bf\,I=\frac{a}{2}$

Answer 2

Answer:

Hope it will help you ☺️