JEE Mains · Maths · STD 12 - 7.2 definite integral
Let \(f ( x )=\int \frac{\sqrt{ x }}{(1+ x )^{2}} d x ( x \geq 0) .\) Then \(f (3)- f (1)\) is equal to
- A \(-\frac{\pi}{6}+\frac{1}{2}+\frac{\sqrt{3}}{4}\)
- B \(\frac{\pi}{6}+\frac{1}{2}-\frac{\sqrt{3}}{4}\)
- C \(-\frac{\pi}{12}+\frac{1}{2}+\frac{\sqrt{3}}{4}\)
- D \(\frac{\pi}{12}+\frac{1}{2}-\frac{\sqrt{3}}{4}\)
Answer & Solution
Correct Answer
(D) \(\frac{\pi}{12}+\frac{1}{2}-\frac{\sqrt{3}}{4}\)
Step-by-step Solution
Detailed explanation
\(f(x)=\int_{1}^{3} \frac{\sqrt{x} d x}{(1+x)^{2}}=\int_{1}^{\sqrt{3}} \frac{t \cdot 2 t d t}{\left(1+t^{2}\right)^{2}} \quad(\) put \(\sqrt{x}=t)\) \(=\left(-\frac{t}{1+t^{2}}\right)_{1}^{\sqrt{3}}+\left(\tan ^{-1} t\right)_{1}^{\sqrt{3}} \quad[\) Appling by parts \(]\)…
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 the function \(f(x)=\left\{\begin{array}{ll}k_{1}(x-\pi)^{2}-1, & x \leq \pi \\ k_{2} \cos x, & x>\pi\end{array}\right.\) is twice differentiable, then the ordered pair \(\left( k _{1}, k _{2}\right)\) is equal toJEE Mains 2020 Hard
- The area (in square units) bounded by the curves \(y = \sqrt x \) and \(2y - x + 3 = 0\) and \(X-\) axis and lying in the first quadrant is :JEE Mains 2013 Medium
- Let \(S=\left\{z \in C : z^{2}+\bar{z}=0\right\}\). Then \(\sum \limits_{z \in S}(\operatorname{Re}(z)+\operatorname{Im}(z))\) is equal to\(......\)JEE Mains 2022 Hard
- Let the plane passing through the point \((-1,0,-2)\) and perpendicular to each of the planes \(2 x+y-\) \(z=2\) and \(x-y-z=3\) be \(a x+b y+c z+8=0\). then the value of \(a+b+c\) is equal to:JEE Mains 2021 Medium
- If \(\int {{x^5}\,{e^{ - {x^2}}}\,dx\, = \,g\,(x)\,{e^{ - {x^2}}} + \,c,} \) where \(c\) is a constant of integration, then \(g(-1)\) is equal toJEE Mains 2019 Hard
- Let \(\overrightarrow{ x }\) be a vector in the plane containing vectors \(\overrightarrow{ a }=2 \hat{ i }-\hat{ j }+\hat{ k }\) and \(\overrightarrow{ b }=\hat{ i }+2 \hat{ j }-\hat{ k }\). If the vector \(\overrightarrow{ x }\) is perpendicular to \((3 \hat{ i }+2 \hat{ j }-\hat{ k })\) and its projection on \(\overrightarrow{ a }\) is \(\frac{17 \sqrt{6}}{2},\) then the value of \(|\overrightarrow{ x }|^{2}\) is equal to ...... .JEE Mains 2021 Medium
More PYQs from JEE Mains
- If the function \(f\left( x \right) = \left\{ {\begin{array}{*{20}{c}}
{\frac{{\sqrt {2 + \cos \,x} - 1}}{{\left( {\pi - {x^2}} \right)}},}&{x \ne \pi } \\
{k\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,,}&{x = \pi }
\end{array}} \right.\) is continuous at \(x\, =\pi \) , then \(k\) equalsJEE Mains 2014 Hard - A point on the ellipse, \(4x^2 + 9y^2 = 36\), where the normal is parallel to the line, \(4x -2y-5 = 0\) , isJEE Mains 2013 Hard
- If the point \((1, 4)\) lies inside the circle \(x^2 + y^2-6x - 10y + p = 0\) and the circle does not touch or intersect the coordinate axes, then the set of all possible values of \(p\) is the intervalJEE Mains 2014 Hard
- The sides of a rhombus \(ABCD\) are parallel to the lines, \(x - y + 2\, = 0\) and \(7x - y + 3\, = 0\). If the diagonals of the rhombus intersect at \(P( 1, 2)\) and the vertex \(A\) ( different from the origin) is on the \(y\) axis, then the ordinate of \(A\) isJEE Mains 2018 Hard
- In an examination,\(5\) students have been allotted their seats as per their roll numbers. The number of ways, in which none of the students sits on the allotted seat, is \(..........\).JEE Mains 2023 Hard
- Let \(f( x)\) be a polynomial of degree four having extreme values at \( x=1 \) and \( x=2\) . If \(\mathop {\lim }\limits_{x \to 0} \left[ {1 + \frac{{f\left( x \right)}}{{{x^2}}}} \right] = 3\),then \(f\left( 2 \right)\) is equal to :JEE Mains 2015 Hard