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Q)

Evaluate the following problems using properties of integration: $\int\limits_{0}^{3}\large\frac{\sqrt{x}dx}{\sqrt{x}+\sqrt{3-x}}$

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A)
Toolbox:
  • $\int \limits_0^a f(x) dx=\int \limits_0^a f(a-x) dx$
$\int\limits_{0}^{3}\large\frac{\sqrt{x}dx}{\sqrt{x}+\sqrt{3-x}}$
Step 1:
$\int\limits_{0}^{3}\large\frac{\sqrt{x}dx}{\sqrt{x}+\sqrt{3-x}}$$=I$
$\qquad=\int\limits_{0}^{3}\large\frac{\sqrt{3-x}dx}{\sqrt{3-x}+\sqrt{3-(3-x)}}$
$\qquad=\int\limits_{0}^{3}\large\frac{\sqrt{3-x}dx}{\sqrt{3-x}+\sqrt{x}}$
Step 2:
$\therefore 2I=\int \limits _0^3 \bigg[\large\frac{\sqrt x}{\sqrt x +\sqrt 3 -x}+\frac{\sqrt {3-x}}{\sqrt {3-x}+\sqrt x}\bigg]$$dx$
$\qquad=\int \limits_0^3 dx=x \bigg]_0^3=3$
$=>I=\large\frac{3}{2}$

 

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