Definite Integration36 questions

Definite Integration — JEE Maths Practice Questions & Solutions

36 questions on Definite Integration with full step-by-step solutions, including past-year (PYQ) problems. Free to practice.

hard
The value of ∫0n2[x ] dx\displaystyle\int_{0}^{n^{2}}\big[\sqrt{x}\,\big]\,dx, where [ ⋅ ][\,\cdot\,] denotes the greatest integer function and n∈Nn\in\mathbb{N}, is
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hard
The value of ∫−11{x2+x−3} dx\displaystyle\int_{-1}^{1}\{x^{2}+x-3\}\,dx, where { ⋅ }\{\,\cdot\,\} denotes the fractional part of xx, is
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hard
The value of ∫−2n 2n+12{x2}sin⁡πx dx\displaystyle\int_{-2n}^{\,2n+\frac12}\left\{\dfrac{x}{2}\right\}\sin\pi x\,dx, where { ⋅ }\{\,\cdot\,\} denotes the fractional part, is
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hard
Let In=∫0∞e−xsin⁡nx dxI_{n}=\displaystyle\int_{0}^{\infty}e^{-x}\sin^{n}x\,dx, n∈Nn\in\mathbb{N}, n>1n>1. Then I2008I2006\dfrac{I_{2008}}{I_{2006}} equals
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hard
The value of the integral ∣ ∫02π[2sin⁡x] dx ∣\left|\,\displaystyle\int_{0}^{2\pi}[2\sin x]\,dx\,\right|, where [ ⋅ ][\,\cdot\,] denotes the greatest integer function, is
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hard
If I=∫0π/2e−αsin⁡x dxI=\displaystyle\int_{0}^{\pi/2}e^{-\alpha\sin x}\,dx, α∈(0,∞)\alpha\in(0,\infty), then
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medium
The value of ∫eπ2[log⁡πx] d(log⁡ex)\displaystyle\int_{e}^{\pi^{2}}\big[\log_{\pi}x\big]\,d(\log_{e}x), where [ ⋅ ][\,\cdot\,] denotes the greatest integer function, is
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medium
∫0π/4(πx−4x2)ln⁡(1+tan⁡x) dx\displaystyle\int_{0}^{\pi/4}\big(\pi x-4x^{2}\big)\ln(1+\tan x)\,dx equals
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medium
∫0π/2sin⁡x sin⁡2x sin⁡3x sin⁡4x dx\displaystyle\int_{0}^{\pi/2}\sin x\,\sin 2x\,\sin 3x\,\sin 4x\,dx equals
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medium
If f(x)=∫01dt1+∣x−t∣f(x)=\displaystyle\int_{0}^{1}\dfrac{dt}{1+|x-t|}, then the value of f′ ⁣(12)f'\!\left(\dfrac12\right) is
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medium
Let f(x)f(x) be the maximum and g(x)g(x) be the minimum of {x∣x∣, x2∣x∣}\big\{x|x|,\,x^{2}|x|\big\}. Then ∫−11[f(x)−g(x)] dx\displaystyle\int_{-1}^{1}\big[f(x)-g(x)\big]\,dx equals
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medium
Let T>0T>0 be a fixed real number. Suppose ff is continuous with f(x+T)=f(x)f(x+T)=f(x) for all x∈Rx\in\mathbb{R}. If I=∫0Tf(x) dxI=\displaystyle\int_{0}^{T}f(x)\,dx, then ∫33+3Tf(2x) dx=KI\displaystyle\int_{3}^{3+3T}f(2x)\,dx=KI, where KK is
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medium
lim⁡n→∞(cos⁡π2n cos⁡2π2n cos⁡3π2n⋯cos⁡(n−1)π2n)1/n\displaystyle\lim_{n\to\infty}\left(\cos\dfrac{\pi}{2n}\,\cos\dfrac{2\pi}{2n}\,\cos\dfrac{3\pi}{2n}\cdots\cos\dfrac{(n-1)\pi}{2n}\right)^{1/n} equals
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medium
If [ ⋅ ][\,\cdot\,] denotes the greatest integer function, ∫02π[ ∣sin⁡x∣−∣cos⁡x∣ ] dx\displaystyle\int_{0}^{2\pi}\big[\,|\sin x|-|\cos x|\,\big]\,dx equals
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medium
∫−22max⁡(x+∣x∣, x−[x]) dx\displaystyle\int_{-2}^{2}\max\big(x+|x|,\ x-[x]\big)\,dx, where [x][x] denotes the greatest integer ≤x\le x, equals
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medium
∫02πx ln⁡ ⁣(3+cos⁡x3−cos⁡x)dx\displaystyle\int_{0}^{2\pi}x\,\ln\!\left(\dfrac{3+\cos x}{3-\cos x}\right)dx equals
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medium
The range of the function f(x)=∫1x∣t∣ dtf(x)=\displaystyle\int_{1}^{x}|t|\,dt, x∈[−12,12]x\in\left[-\dfrac12,\dfrac12\right], is
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medium
The value of the definite integral ∫−1/21/2(sin⁡−1(3x−4x3)−cos⁡−1(4x3−3x))dx\displaystyle\int_{-1/2}^{1/2}\Big(\sin^{-1}(3x-4x^{3})-\cos^{-1}(4x^{3}-3x)\Big)dx is
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medium
A function g(x)g(x) is given by g(x)=x1+(log⁡ex)(log⁡ex)⋯∞g(x)=\dfrac{x}{1+(\log_{e}x)(\log_{e}x)\cdots\infty}, where x∈[1,∞)x\in[1,\infty). Then ∫12eg(x) dx\displaystyle\int_{1}^{2e}g(x)\,dx is equal to
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medium
Suppose f(x)=∫0x1+t4 dtf(x)=\displaystyle\int_{0}^{x}\sqrt{1+t^{4}}\,dt for all real xx and let f(1)=cf(1)=c. Then the value of (f−1)′(c)\big(f^{-1}\big)'(c) is
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medium
If xx satisfies (∫01dtt2+2tcos⁡α+1)x2−(∫−33t2sin⁡2tt2+1 dt)x−2=0\left(\displaystyle\int_{0}^{1}\dfrac{dt}{t^{2}+2t\cos\alpha+1}\right)x^{2}-\left(\displaystyle\int_{-3}^{3}\dfrac{t^{2}\sin 2t}{t^{2}+1}\,dt\right)x-2=0 for 0<α<π0<\alpha<\pi, then the value of xx is
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medium
A cubic f(x)f(x) vanishes at x=−2x=-2 and has local minimum/maximum at x=−1x=-1 and x=13x=\dfrac13. If ∫−11f(x) dx=143\displaystyle\int_{-1}^{1}f(x)\,dx=\dfrac{14}{3}, then f(1)=f(1)=
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medium
If A=∫1sin⁡θt dt1+t2A=\displaystyle\int_{1}^{\sin\theta}\dfrac{t\,dt}{1+t^{2}} and B=∫1cosec⁡θdtt (1+t2)B=\displaystyle\int_{1}^{\operatorname{cosec}\theta}\dfrac{dt}{t\,(1+t^{2})}, then ∣AA2BeA+BB2−11A2+B2−1∣\begin{vmatrix}A & A^{2} & B\\[2pt] e^{A+B} & B^{2} & -1\\[2pt] 1 & A^{2}+B^{2} & -1\end{vmatrix} equals
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medium
If f(x)f(x) is a monotonic and differentiable function, then ∫f(a)f(b)2x(b−f−1(x)) dx\displaystyle\int_{f(a)}^{f(b)}2x\big(b-f^{-1}(x)\big)\,dx equals
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medium
If A=∫0πcos⁡x(x+2)2 dxA=\displaystyle\int_{0}^{\pi}\dfrac{\cos x}{(x+2)^{2}}\,dx, then ∫0π/2sin⁡2xx+1 dx\displaystyle\int_{0}^{\pi/2}\dfrac{\sin 2x}{x+1}\,dx is equal to
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medium
The value of ∫0nπ+t(∣cos⁡x∣+∣sin⁡x∣) dx\displaystyle\int_{0}^{n\pi+t}\big(|\cos x|+|\sin x|\big)\,dx, where 0<t<π20<t<\dfrac{\pi}{2}, is
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medium
If ∫01e5x6(x−α) dx=0\displaystyle\int_{0}^{1}e^{5x^{6}}(x-\alpha)\,dx=0, then
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medium
The value of the integral ∫01/3dx(1+x2)1−x2\displaystyle\int_{0}^{1/\sqrt3}\dfrac{dx}{(1+x^{2})\sqrt{1-x^{2}}} must be
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medium
∫0∞f ⁣(xn+x−n)ln⁡x dxx\displaystyle\int_{0}^{\infty}f\!\big(x^{n}+x^{-n}\big)\ln x\,\dfrac{dx}{x} equals
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medium
The value of ∫−13(tan⁡−1xx2+1+tan⁡−1x2+1x)dx\displaystyle\int_{-1}^{3}\left(\tan^{-1}\dfrac{x}{x^{2}+1}+\tan^{-1}\dfrac{x^{2}+1}{x}\right)dx is
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medium
If f(x)=∫1xdt2+t4f(x)=\displaystyle\int_{1}^{x}\dfrac{dt}{2+t^{4}}, then
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medium
∫0π/4(xxsin⁡x+cos⁡x)2dx\displaystyle\int_{0}^{\pi/4}\left(\dfrac{x}{x\sin x+\cos x}\right)^{2}dx equals
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easy
∫0π/2sin⁡x1+cos⁡x+sin⁡x dx\displaystyle\int_{0}^{\pi/2}\dfrac{\sin x}{1+\cos x+\sin x}\,dx equals
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easy
If f(x)=∫1/xxcos⁡t2 dtf(x)=\displaystyle\int_{1/x}^{\sqrt{x}}\cos t^{2}\,dt for x>0x>0, then df(x)dx\dfrac{df(x)}{dx} is
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easy
If ∫0π/2sin⁡10x cos⁡10x dx=2−n∫0π/2(sin⁡10x+cos⁡10x) dx\displaystyle\int_{0}^{\pi/2}\sin^{10}x\,\cos^{10}x\,dx=2^{-n}\displaystyle\int_{0}^{\pi/2}\big(\sin^{10}x+\cos^{10}x\big)\,dx, n∈Nn\in\mathbb{N}, then n=n=
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easy
If ∫sin⁡x1t2f(t) dt=1−sin⁡x\displaystyle\int_{\sin x}^{1}t^{2}f(t)\,dt=1-\sin x for all x∈(0,π2]x\in\left(0,\dfrac{\pi}{2}\right], then the value of f ⁣(13)f\!\left(\dfrac{1}{\sqrt3}\right) is
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