JEE Main 2021 · MathsMultiple choiceSingle correctHardMulti-step

JEE Main 27 August 2021, Shift 2, Maths Q70

Question 70 of 90 in this shift, Maths question 10 of 30, Section A.

The value of the integral

\int_0^1 \frac{\sqrt{x}\, dx}{(1+x)(1+3x)(3+x)}

is :

(1) $\frac{\pi}{8}\left(1 - \frac{\sqrt{3}}{2}\right)$ Official answer
(2) $\frac{\pi}{8}\left(1 - \frac{\sqrt{3}}{6}\right)$
(3) $\frac{\pi}{4}\left(1 - \frac{\sqrt{3}}{6}\right)$
(4) $\frac{\pi}{4}\left(1 - \frac{\sqrt{3}}{2}\right)$

Official answer

Option 1

NTA final key.

Chapter: Integrals
Topic: Integration by Substitution
Idea tested: Fractional Powers LCM Substitution

Same idea in other shifts

Asked 5× in all

8 Apr 2024, Shift 2 · Q26If $\int \frac{1}{\sqrt[5]{(x-1)^4(x+3)^6}}\,dx = A\left(\frac{\alpha x - 1}{\beta x + 3}\right)^B + C$ , where C is the constant of…HardNumerical value
31 Jan 2024, Shift 1 · Q27If the integral $525\int_0^{\frac{\pi}{2}}\sin2x\cos^{\frac{11}{2}}x\left(1+\cos^{\frac{5}{2}}x\right)^{\frac{1}{2}}dx$ is equal to…HardNumerical value
12 Apr 2023, Shift 1 · Q27Let $I(x)=\int\sqrt{\frac{x+7}{x}}\,dx$ and $I(9)=12+7\log_e 7$ . If $I(1)=\alpha+7\log_e(1+2\sqrt{2})$ , then $\alpha^4$ is equal to…HardNumerical value
4 Sep 2020, Shift 1 · Q62Let $f(x) = \int \frac{\sqrt{x}}{(1+x)^2}\,\mathrm{d}x$ $(x \ge 0)$ . Then $f(3) - f(1)$ is equal to :MediumSingle correct

Question text from the official JEE Main paper published by NTA; answer from the final answer key. Chapter, topic and idea tags, difficulty and skill are Lumi’s. Lumi analyses 108 of the 174 JEE Main shifts held from 2017 to 2026.