SATHEE: Definite Integration Question 59

Definite Integration Question 59

Question 59

The value of $\int_{2}^{3} \frac{\sqrt{x}}{\sqrt{5 - x} + \sqrt{x}} d x$ is …… .

$(1994, 2 M)$

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Solution:

Let $I = \int_{2}^{3} \frac{\sqrt{x}}{\sqrt{5 - x} + \sqrt{x}} d x$

$\begin{aligned} \Rightarrow I = \int_{2}^{3} \frac{\sqrt{2 + 3 - x}}{\sqrt{(2 + 3) - (5 - x)} + \sqrt{2 + 3 - x}} d x & \Rightarrow I = \int_{2}^{3} \frac{\sqrt{5 - x}}{\sqrt{x} + \sqrt{5 - x}} d x \end{aligned}$

On adding Eqs. (i) and (ii), we get

$2 I = \int_{2}^{3} \frac{\sqrt{x} + \sqrt{5 - x}}{\sqrt{5 - x} + \sqrt{x}} d x \Rightarrow 2 I = \int_{2}^{3} 1 d x = 1 \Rightarrow I = \frac{1}{2}$

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Definite Integration Question 59

Question 59

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