Indefinite Integration: The integral intfracdxsqrt(2x+5)-sqrt(2x+3) is equal to (where C is… | JEE Main 2021

JEE Main 2021 (Online) 3rd August Evening ShiftQuestion 1

The integral $\int\frac{dx}{\sqrt{2x+5}-\sqrt{2x+3}}$ is equal to (where $C$ is a constant of integration)

Indefinite Integration

$\frac{1}{3}\left[(2x+5)^{\frac{3}{2}}+(2x+3)^{\frac{3}{2}}\right]+C$$\frac{1}{6}\left[(2x+5)^{\frac{2}{3}}+(2x+3)^{\frac{2}{3}}\right]+C$$\frac{1}{6}\left[(2x+5)^{\frac{3}{2}}+(2x+3)^{\frac{3}{2}}\right]+C$✓ Correct answer$\frac{1}{12}\left[(2x+5)^{\frac{2}{3}}+(2x+3)^{\frac{2}{3}}\right]+C$

Solution

Concept. Rationalising gives half the sum of two square roots. Formulas used. (1) $\dfrac1{\sqrt{2x+5}-\sqrt{2x+3}}=\dfrac{\sqrt{2x+5}+\sqrt{2x+3}}{(2x+5)-(2x+3)}=\dfrac{\sqrt{2x+5}+\sqrt{2x+3}}2$(2) $\int\sqrt{2x+c}\,dx=\frac13(2x+c)^{3/2}$Step 1 - Integrate. By (1), (2), $$I=\frac12\cdot\frac13\Big[(2x+5)^{3/2}+(2x+3)^{3/2}\Big]+C=\frac16\Big[(2x+5)^{3/2}+(2x+3)^{3/2}\Big]+C.$$ Answer: (C)

Indefinite Integration - JEE Mathematics question (JEE Main 2021 (Online) 3rd August Evening Shift)

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