Mathematics

Evaluate $$\displaystyle\int_{2}^{3}{\dfrac{\sqrt{x}dx}{\sqrt{5-x}+\sqrt{x}}}$$.


SOLUTION
Let, $$I=\displaystyle\int_{2}^{3}{\dfrac{\sqrt{x}dx}{\sqrt{5-x}+\sqrt{x}}}$$        .......$$(1)$$

We have $$\displaystyle\int_{a}^{b}{f\left(x\right)dx}=\displaystyle\int_{a}^{b}{f\left(a+b-x\right)dx}$$

$$\Rightarrow I=\displaystyle\int_{2}^{3}{\dfrac{\sqrt{x}dx}{\sqrt{5-x}+\sqrt{x}}}$$

Replace $$x\rightarrow 3+2-x=5-x$$

$$I=\displaystyle\int_{2}^{3}{\dfrac{\sqrt{5-x}dx}{\sqrt{5-\left(5-x\right)}+\sqrt{5-x}}}$$

$$I=\displaystyle\int_{2}^{3}{\dfrac{\sqrt{5-x}dx}{\sqrt{x}+\sqrt{5-x}}}$$     .......$$(2)$$

Adding $$(1)$$ and $$(2)$$ we get

$$2I=\displaystyle\int_{2}^{3}{\dfrac{\sqrt{x}dx}{\sqrt{5-x}+\sqrt{x}}}+\displaystyle\int_{2}^{3}{\dfrac{\sqrt{5-x}dx}{\sqrt{x}+\sqrt{5-x}}}$$

$$2I=\displaystyle\int_{2}^{3}{\dfrac{\left(\sqrt{x}+\sqrt{5-x}\right)dx}{\sqrt{5-x}+\sqrt{x}}}$$

$$\Rightarrow 2I=\displaystyle\int_{2}^{3}{dx}$$

$$\Rightarrow 2I=\left[x\right]_{2}^{3}$$

$$\Rightarrow 2I=3-2=1$$

$$\therefore I=\dfrac{1}{2}$$
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Subjective Medium Published on 17th 09, 2020
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