CUET UG 2025 — Mathematics Calculus

Given:

$$\int_2^3 \frac{\sqrt{x}}{\sqrt{x} + \sqrt{5-x}}dx$$∫23​x​+5−x​x​​dx

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The King Property of definite integrals states:

$$\int_a^b f(x)dx = \int_a^b f(a+b-x)dx$$∫ab​f(x)dx=∫ab​f(a+b−x)dx

Here $a = 2$a=2 and $b = 3$b=3, so $a + b = 5$a+b=5.

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Let $\displaystyle I = \int_2^3 \frac{\sqrt{x}}{\sqrt{x} + \sqrt{5-x}}dx$I=∫23​x​+5−x​x​​dx ... (1)

Replacing $x$x with $(5-x)$(5−x):

$$I = \int_2^3 \frac{\sqrt{5-x}}{\sqrt{5-x} + \sqrt{5-(5-x)}}dx$$I=∫23​5−x​+5−(5−x)​5−x​​dx

$\displaystyle I = \int_2^3 \frac{\sqrt{5-x}}{\sqrt{5-x} + \sqrt{x}}dx$I=∫23​5−x​+x​5−x​​dx ... (2)

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Adding equations (1) and (2):

$$I + I = \int_2^3 \frac{\sqrt{x}}{\sqrt{x} + \sqrt{5-x}}dx + \int_2^3 \frac{\sqrt{5-x}}{\sqrt{x} + \sqrt{5-x}}dx$$I+I=∫23​x​+5−x​x​​dx+∫23​x​+5−x​5−x​​dx

$$2I = \int_2^3 \frac{\sqrt{x} + \sqrt{5-x}}{\sqrt{x} + \sqrt{5-x}}dx$$2I=∫23​x​+5−x​x​+5−x​​dx

$$2I = \int_2^3 1 \, dx$$2I=∫23​1dx

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$$2I = [x]_2^3$$2I=[x]23​

$$2I = 3 - 2$$2I=3−2

$$2I = 1$$2I=1

$$I = \frac{1}{2}$$I=21​

Therefore, the value of the integral is $\frac{1}{2}$21​.