Definite Integration Question 59
Question 59
- The value of $\int_{2}^{3} \frac{\sqrt{x}}{\sqrt{5-x}+\sqrt{x}} d x$ is …… .
$(1994,2 \mathrm{M})$
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Solution:
- Let $I=\int_{2}^{3} \frac{\sqrt{x}}{\sqrt{5-x}+\sqrt{x}} d x$
$$ \begin{aligned} & \Rightarrow \quad I=\int_{2}^{3} \frac{\sqrt{2+3-x}}{\sqrt{(2+3)-(5-x)}+\sqrt{2+3-x}} d x \ & \Rightarrow \quad I=\int_{2}^{3} \frac{\sqrt{5-x}}{\sqrt{x}+\sqrt{5-x}} d x \end{aligned} $$
On adding Eqs. (i) and (ii), we get
$$ 2 I=\int_{2}^{3} \frac{\sqrt{x}+\sqrt{5-x}}{\sqrt{5-x}+\sqrt{x}} d x \Rightarrow 2 I=\int_{2}^{3} 1 d x=1 \Rightarrow I=\frac{1}{2} $$
