JEE Mains · Maths · STD 12 - 7.2 definite integral
The value of the integral \(\int \limits_{0}^{1} \frac{\sqrt{x} d x}{(1+x)(1+3 x)(3+x)}\) is:
- A \(\frac{\pi}{8}\left(1-\frac{\sqrt{3}}{2}\right)\)
- B \(\frac{\pi}{4}\left(1-\frac{\sqrt{3}}{6}\right)\)
- C \(\frac{\pi}{8}\left(1-\frac{\sqrt{3}}{6}\right)\)
- D \(\frac{\pi}{4}\left(1-\frac{\sqrt{3}}{2}\right)\)
Answer & Solution
Correct Answer
(A) \(\frac{\pi}{8}\left(1-\frac{\sqrt{3}}{2}\right)\)
Step-by-step Solution
Detailed explanation
\(I=\int_{0}^{1} \frac{\sqrt{x}}{(1+x)(1+3 x)(3+x)} \,d x\) Let \(\mathrm{x}=\mathrm{t}^{2} \Rightarrow \mathrm{dx}=2 \mathrm{t} \mathrm{dt}\)…
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- If \(\int {{x^5}{e^{ - 4{x^3}}}\,dx = \frac{1}{{48}}{e^{ - 4{x^3}}}f\left( x \right) + C} \), where \(C\) is a constant of integration, then \(f(x)\) is equal toJEE Mains 2019 Hard
- Let \(f\) : \(A \to B\) be a function defined as \(f(x)\, = \frac{{x - 1}}{{x - 2}}\) , where \(A\, = R - \{2\}\) and \(B\, = R - \{1\}\) . Then \(f\) isJEE Mains 2018 Hard
- Let \(g\) be a differentiable function such that \(\int_0^x g(t) d t=x-\int_0^x \operatorname{tg}(t) d t, x \geq 0\) and let \(y=y(x)\) satisfy the differential equation \(\frac{d y}{d x}-y \tan x=\) \(2(x+1) \sec x g(x), x \in\left[0, \frac{\pi}{2}\right)\). If \(y(0)=0\), then \(y\left(\frac{\pi}{3}\right)\) is equal toJEE Mains 2025 Medium
- Let the sum of the first \(n\) terms of an A.P. be \(3n^2 + 5n\). Then the sum of squares of the first \(10\) terms of the A.P. is:JEE Mains 2026 Medium
- Let \(A = \left\{ {\left( {x,y} \right):{y^2} \le 4x,y - 2x \ge - 4} \right\}\) .The area of the region \(A\) isJEE Mains 2014 Hard
- Consider the following three statements for the function \( f:(0,\infty)\rightarrow\mathbb{R} \) defined by
\( f(x)=|log_{e}x|-|x-1| \):
(I) f is differentiable at all \( x>0 \).
(II) f is increasing in (0, 1).
(III) f is decreasing in (1, ∞).
Then:JEE Mains 2026 Hard
More PYQs from JEE Mains
- Consider the statistics of two sets of observations as follows :
If the variance of the combined set of these two observations is \(\frac{17}{9},\) then the value of \(n\) is equal to ..... .Size Mean Variance Observation \(I\) \(10\) \(2\) \(2\) Observation \(II\) \(n\) \(3\) \(1\) JEE Mains 2021 Hard - If the variance of the following frequency distribution is \(50\) then \(x\) is equal to:
Class \(10-20\) \(20-30\) \(30-40\) Frequency \(2\) \(x\) \(2\) JEE Mains 2020 Medium - If \(\mathrm{n}\) is the number of ways five different employees can sit into four indistinguishable offices where any office may have any number of persons including zero, then \(\mathrm{n}\) is equal to :JEE Mains 2024 Medium
- Let \(y(x)=(1+x)\left(1+x^2\right)\left(1+x^4\right)\left(1+x^8\right)\left(1+x^{16}\right)\) Then \(y^{\prime}-y^{\prime \prime}\) at \(x=-1\) is equal toJEE Mains 2023 Hard
- Let \(z_0\) be a root of the quadratic equation, \(x^2 + x + 1= 0.\) If \(z = 3 + \,6iz_0^{81}\, - 3iz_0^{93}, \) then arg \(z\) is equal toJEE Mains 2019 Hard
- If \(a \in R\) and the equation \( - 3{\left( {x - \left[ x \right]} \right)^2} + 2\left( {x - \left[ x \right]} \right) + {a^2} = 0\) (where \([x]\) denotes the greatest integer \(\leq\,x\))has no integral solution ,then all possible values of \(a\) lie in the intervalJEE Mains 2014 Hard