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Friday, 9 October

JEE Main Maths · Class 12

Integrals: JEE Main previous year questions

Integrals had 73 questions in the 42 JEE Main shifts Lumi analysed, about 1.6 a shift, asked in 42 of 42 shifts. It ranks 4 of 22 Maths chapters by questions; 53% of its questions were hard.

Questions
73
in 42 of 42 shifts
A shift, on average
1.6
of 25 Maths questions
Since 2024
+0.2
1.5 a shift in 2017–23, 1.7 in 2024–26
Numerical answer
22
30% of its questions

Year by year

How often it came

Questions in the analysed shifts of each year. Years differ in how many shifts Lumi analysed, so read the bars with the shift count.

Questions each year
09182’174’209’216’227’2318’2413’2514’26

2017: 1 shifts · 2020: 4 shifts · 2021: 4 shifts · 2022: 4 shifts · 2023: 4 shifts · 2024: 8 shifts · 2025: 8 shifts · 2026: 9 shifts

Difficulty by year
  • 20172
  • 20204
  • 20219
  • 20226
  • 20237
  • 202418
  • 202513
  • 202614
  • Easy
  • Medium
  • Hard

Inside the chapter

Topics and the ideas that repeat

8 topics and 37 distinct ideas; 14 ideas were asked more than once. Most questions are single correct and multi-step.

Topics, by questions

8 questions had no close topic in our taxonomy and are counted for the chapter only.

Ideas asked most
14 repeat

Study it on Lumi

Integrals: the interactive lesson

Every question

All 73 Integrals questions

Newest first. Each opens with its options and the official answer.

2026 14 questions

  1. 2 Apr 2026, Shift 1 · Q19Let [⋅][\cdot] denote the greatest integer function. Then the value of ∫03(ex+e−x[x]!)dx\int_0^3\left(\frac{e^x+e^{-x}}{[x]!}\right)dx is :MediumSingle correct
  2. 2 Apr 2026, Shift 1 · Q25If α=∫023log⁡2(x2+4)dx+∫242x−4 dx\alpha=\int_0^{2\sqrt3}\log_2\left(x^2+4\right)dx+\int_2^4\sqrt{2^x-4}\,dx, then α2\alpha^2 is equal to __________.HardNumerical value
  3. 2 Apr 2026, Shift 2 · Q17The value of ∫020π(sin⁡4x+cos⁡4x)dx\int_0^{20\pi} (\sin^4 x + \cos^4 x)dx is equal to:EasySingle correct
  4. 2 Apr 2026, Shift 2 · Q19Let f(x)=∫16x+24x2+2x−15dxf(x) = \int\frac{16x + 24}{x^2 + 2x - 15}dx. If f(4)=14log⁡6(3)f(4) = 14\log_6(3) and…MediumSingle correct
  5. 4 Apr 2026, Shift 1 · Q19Let ∫−22(∣sin⁡x∣+[xsin⁡x])dx=2(3−cos⁡2)+β\int_{-2}^{2}\left(|\sin x|+[x\sin x]\right)dx=2(3-\cos 2)+\beta, where [⋅][\cdot] is the greatest integer function. Then…HardSingle correct
  6. 4 Apr 2026, Shift 2 · Q4If α=1\alpha=1 and β=1+i2\beta=1+i\sqrt{2}, where i=−1i=\sqrt{-1} are two roots of the equation x3+ax2+bx+c=0x^3+ax^2+bx+c=0, a,b,c∈Ra,b,c\in\mathbb{R}, then…MediumSingle correct
  7. 4 Apr 2026, Shift 2 · Q20The integral ∫01cot⁡−1(1+x+x2)dx\int_0^1\cot^{-1}\left(1+x+x^2\right)dx is equal to:HardSingle correct
  8. 5 Apr 2026, Shift 1 · Q18The value of the integral ∫0∞log⁡e(x)x2+4 dx\int_0^{\infty} \frac{\log_e(x)}{x^2+4}\,dx is:HardSingle correct
  9. 5 Apr 2026, Shift 1 · Q20The value of the integral ∫π/6π/3(4−cosec⁡2xcos⁡4x)dx\int_{\pi/6}^{\pi/3}\left(\frac{4 - \operatorname{cosec}^2 x}{\cos^4 x}\right)dx is:MediumSingle correct
  10. 5 Apr 2026, Shift 2 · Q18Let (21−a+21+a)(2^{1-a}+2^{1+a}), f(a)f(a), (3a+3−a)(3^a+3^{-a}) be in A.P. and α\alpha be the minimum value of f(a)f(a). Then the value of the integral…HardSingle correct
  11. 6 Apr 2026, Shift 1 · Q18The value of the integral ∫−π4π4(32cos⁡4x1+esin⁡x)dx\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}}\left(\frac{32\cos^4 x}{1+e^{\sin x}}\right)dx is:MediumSingle correct
  12. 6 Apr 2026, Shift 2 · Q20The value of the integral ∫−11(x3+∣x∣+1x2+2∣x∣+1)dx\int_{-1}^{1}\left(\frac{x^3+|x|+1}{x^2+2|x|+1}\right)dx is equal to :MediumSingle correct
  13. 8 Apr 2026, Shift 2 · Q16The value of the integral ∫02x (x2+x+1)(x+1)(x4+x2+1) dx\int_0^2 \frac{\sqrt{x}\,(x^2 + x + 1)}{(\sqrt{x} + 1)\left(\sqrt{x^4 + x^2 + 1}\right)}\,dx is equal to :HardSingle correct
  14. 8 Apr 2026, Shift 2 · Q22If…HardNumerical value

2025 13 questions

  1. 2 Apr 2025, Shift 1 · Q23Let [⋅][\cdot] denote the greatest integer function. If ∫0e3[1ex−1]dx=α−log⁡e2\int_0^{e^3}\left[\frac{1}{e^{x-1}}\right]dx=\alpha-\log_e2, then α3\alpha^3 is…MediumNumerical value
  2. 2 Apr 2025, Shift 2 · Q184∫01(13+x2+1+x2)dx−3log⁡e(3)4\int_0^1 \left(\frac{1}{\sqrt{3+x^2}+\sqrt{1+x^2}}\right)\mathrm{d}x - 3\log_e(\sqrt{3}) is equal to :HardSingle correct
  3. 2 Apr 2025, Shift 2 · Q19Let (a,b)(a, b) be the point of intersection of the curve x2=2yx^2 = 2y and the straight line y−2x−6=0y - 2x - 6 = 0 in the second quadrant. Then the…MediumSingle correct
  4. 2 Apr 2025, Shift 2 · Q20Let f:[1,∞)→[2,∞)f : [1, \infty) \to [2, \infty) be a differentiable function. If 10∫1xf(t) dt=5xf(x)−x5−910\int_1^x f(t)\,\mathrm{d}t = 5xf(x) - x^5 - 9 for all…MediumSingle correct
  5. 3 Apr 2025, Shift 1 · Q18Let f(x)=∫x33−x2 dxf(x) = \int x^3\sqrt{3 - x^2}\,dx. If 5f(2)=−45f(\sqrt{2}) = -4, then f(1)f(1) is equal toMediumSingle correct
  6. 3 Apr 2025, Shift 1 · Q19Let the domain of the function f(x)=log⁡2log⁡4log⁡6(3+4x−x2)f(x) = \log_2 \log_4 \log_6 (3 + 4x - x^2) be (a,b)(a, b). If…HardSingle correct
  7. 3 Apr 2025, Shift 2 · Q18The integral ∫0π8x dx4cos⁡2x+sin⁡2x\int_0^{\pi} \frac{8x\,dx}{4\cos^2 x + \sin^2 x} is equal toMediumSingle correct
  8. 4 Apr 2025, Shift 2 · Q18Let f(x)+2f(1x)=x2+5f(x)+2f\left(\frac{1}{x}\right)=x^2+5 and 2g(x)−3g(12)=x2g(x)-3g\left(\frac{1}{2}\right)=x, x>0x>0. If α=∫12f(x) dx\alpha=\int_1^2 f(x)\,dx, and…HardSingle correct
  9. 4 Apr 2025, Shift 2 · Q25If…HardNumerical value
  10. 7 Apr 2025, Shift 1 · Q18The integral ∫0π(x+3)sin⁡x1+3cos⁡2x dx\int_0^{\pi} \frac{(x+3)\sin x}{1+3\cos^2 x}\,dx is equal toMediumSingle correct
  11. 7 Apr 2025, Shift 2 · Q24If…HardNumerical value
  12. 8 Apr 2025, Shift 2 · Q18Let f(x)f(x) be a positive function and I1=∫−1212x f(2x(1−2x)) dxI_1 = \int_{-\frac{1}{2}}^{1} 2x\, f\big(2x(1-2x)\big)\,dx and…HardSingle correct
  13. 8 Apr 2025, Shift 2 · Q19The integral ∫−132(∣π2xsin⁡(πx)∣)dx\int_{-1}^{\frac{3}{2}} \left(\left|\pi^2 x \sin(\pi x)\right|\right) dx is equal to :MediumSingle correct

2024 18 questions

  1. 6 Apr 2024, Shift 2 · Q12If ∫1a2sin⁡2x+b2cos⁡2xdx=112tan⁡−1(3tan⁡x)+constant\int\frac{1}{a^2\sin^2x+b^2\cos^2x}dx=\frac{1}{12}\tan^{-1}(3\tan x)+\text{constant}, then the maximum value of asin⁡x+bcos⁡xa\sin x+b\cos x, is :MediumSingle correct
  2. 6 Apr 2024, Shift 2 · Q25Let [t][t] denote the largest integer less than or equal to tt. If…HardNumerical value
  3. 8 Apr 2024, Shift 1 · Q10Let I(x)=∫6sin⁡2x(1−cot⁡x)2dxI(x)=\int\frac{6}{\sin^2x(1-\cot x)^2}dx. If I(0)=3I(0)=3, then I(π12)I\left(\frac{\pi}{12}\right) is equal toMediumSingle correct
  4. 8 Apr 2024, Shift 1 · Q11The value of k∈Nk\in\mathbb{N} for which the integral In=∫01(1−xk)ndxI_n=\int_0^1(1-x^k)^n dx, n∈Nn\in\mathbb{N}, satisfies 147I20=148I21147I_{20}=148I_{21} isHardSingle correct
  5. 9 Apr 2024, Shift 1 · Q8Let ∫2−tan⁡x3+tan⁡x dx=12(αx+log⁡e∣βsin⁡x+γcos⁡x∣)+C\int\frac{2-\tan x}{3+\tan x}\,dx=\frac{1}{2}\left(\alpha x+\log_e|\beta\sin x+\gamma\cos x|\right)+C, where C is the constant of…MediumSingle correct
  6. 9 Apr 2024, Shift 1 · Q28Let…HardNumerical value
  7. 9 Apr 2024, Shift 2 · Q9…HardSingle correct
  8. 9 Apr 2024, Shift 2 · Q11The integral ∫1/43/4cos⁡(2cot⁡−11−x1+x)dx\int_{1/4}^{3/4}\cos\left(2\cot^{-1}\sqrt{\frac{1-x}{1+x}}\right)dx is equal toMediumSingle correct
  9. 9 Apr 2024, Shift 2 · Q12The value of the integral ∫−12log⁡e(x+x2+1)dx\int_{-1}^{2}\log_e\left(x+\sqrt{x^2+1}\right)dx isMediumSingle correct
  10. 29 Jan 2024, Shift 1 · Q11If the value of the integral…HardSingle correct
  11. 29 Jan 2024, Shift 1 · Q12For x∈(−π2,π2)x\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right), if…HardSingle correct
  12. 30 Jan 2024, Shift 2 · Q10Let f:R→Rf: \mathbb{R} \to \mathbb{R} be defined as f(x)=ae2x+bex+cxf(x) = ae^{2x} + be^x + cx. If f(0)=−1f(0) = -1, f′(log⁡e2)=21f'(\log_e 2) = 21 and…MediumSingle correct
  13. 30 Jan 2024, Shift 2 · Q12Let y=f(x)y = f(x) be a thrice differentiable function in (−5,5)(-5, 5). Let the tangents to the curve y=f(x)y = f(x) at (1,f(1))(1, f(1)) and (3,f(3))(3, f(3))…HardSingle correct
  14. 30 Jan 2024, Shift 2 · Q13Let f:R→Rf: \mathbb{R} \to \mathbb{R} be a function defined by f(x)=x(1+x4)1/4f(x) = \frac{x}{(1+x^4)^{1/4}}, and g(x)=f(f(f(f(x))))g(x) = f(f(f(f(x)))). Then,…HardSingle correct
  15. 31 Jan 2024, Shift 1 · Q26Let f:R→Rf:\mathbb{R}\to\mathbb{R} be a function defined by f(x)=4x4x+2f(x)=\frac{4^x}{4^x+2} and…HardNumerical value
  16. 31 Jan 2024, Shift 1 · Q27If the integral 525∫0π2sin⁡2xcos⁡112x(1+cos⁡52x)12dx525\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
  17. 31 Jan 2024, Shift 2 · Q12Let f,g:(0,∞)→Rf,g:(0,\infty)\to\mathbb{R} be two functions defined by f(x)=∫−xx(∣t∣−t2)e−t2dtf(x)=\int_{-x}^{x}\left(|t|-t^2\right)e^{-t^2}dt and…HardSingle correct
  18. 31 Jan 2024, Shift 2 · Q26∣120π3∫0πx2sin⁡xcos⁡xsin⁡4x+cos⁡4xdx∣\left|\frac{120}{\pi^3}\int_0^{\pi}\frac{x^2\sin x\cos x}{\sin^4x+\cos^4x}dx\right| is equal to ____________.HardNumerical value

2023 7 questions

  1. 6 Apr 2023, Shift 1 · Q10Let 5f(x)+4f(1x)=1x+3, x>05f(x)+4f\left(\frac{1}{x}\right)=\frac{1}{x}+3,\ x>0. Then 18∫12f(x) dx18\int_1^2 f(x)\,dx is equal to:MediumSingle correct
  2. 6 Apr 2023, Shift 1 · Q11Let I(x)=∫x2(xsec⁡2x+tan⁡x)(xtan⁡x+1)2 dxI(x)=\int\frac{x^2(x\sec^2x+\tan x)}{(x\tan x+1)^2}\,dx. If I(0)=0I(0)=0, then I(π4)I\left(\frac{\pi}{4}\right) is equal toHardSingle correct
  3. 8 Apr 2023, Shift 2 · Q26Let [t][t] denote the greatest integer function. If ∫02.4[x2] dx=α+β2+γ3+δ5\int_0^{2.4}[x^2]\,dx = \alpha + \beta\sqrt{2} + \gamma\sqrt{3} + \delta\sqrt{5},…MediumNumerical value
  4. 11 Apr 2023, Shift 1 · Q15The value of the integral ∫−log⁡e2log⁡e2ex(log⁡e(ex+1+e2x))dx\int_{-\log_e 2}^{\log_e 2} e^x\left(\log_e\left(e^x + \sqrt{1+e^{2x}}\right)\right)dx is equal toHardSingle correct
  5. 11 Apr 2023, Shift 1 · Q26For m,n>0m, n > 0, let α(m,n)=∫02tm(1+3t)n dt\alpha(m, n) = \int_0^2 t^m(1+3t)^n\,dt. If 11 α(10,6)+18 α(11,5)=p (14)611\,\alpha(10, 6) + 18\,\alpha(11, 5) = p\,(14)^6, then pp is equal…HardNumerical value
  6. 13 Apr 2023, Shift 2 · Q10The value of…HardSingle correct
  7. 13 Apr 2023, Shift 2 · Q26Let fn=∫0π2(∑k=1nsin⁡k−1x)(∑k=1n(2k−1)sin⁡k−1x)cos⁡x dxf_n=\int_0^{\frac{\pi}{2}}\left(\sum_{k=1}^{n}\sin^{k-1}x\right)\left(\sum_{k=1}^{n}(2k-1)\sin^{k-1}x\right)\cos x\,dx,…HardNumerical value

2022 6 questions

  1. 25 Jul 2022, Shift 1 · Q9For any real number xx, let [x][x] denote the largest integer less than equal to xx. Let ff be a real valued function defined on the…HardSingle correct
  2. 25 Jul 2022, Shift 1 · Q27If…MediumNumerical value
  3. 28 Jul 2022, Shift 1 · Q24If ∫0315x31+x2+(1+x2)3 dx=α2+β3\int_0^{\sqrt{3}}\frac{15x^3}{\sqrt{1+x^2+\sqrt{(1+x^2)^3}}}\,dx=\alpha\sqrt{2}+\beta\sqrt{3}, where α,β\alpha,\beta are integers,…HardNumerical value
  4. 24 Jun 2022, Shift 1 · Q26Let f(θ)=sin⁡θ+∫−π/2π/2(sin⁡θ+tcos⁡θ)f(t) dtf(\theta)=\sin\theta+\int_{-\pi/2}^{\pi/2}(\sin\theta+t\cos\theta)f(t)\,dt. Then the value of…HardNumerical value
  5. 24 Jun 2022, Shift 1 · Q27Let max⁡0≤x≤2{9−x25−x}=α\max_{0\le x\le 2}\left\{\frac{9-x^2}{5-x}\right\}=\alpha and min⁡0≤x≤2{9−x25−x}=β\min_{0\le x\le 2}\left\{\frac{9-x^2}{5-x}\right\}=\beta. If…HardNumerical value
  6. 29 Jun 2022, Shift 1 · Q17∫05cos⁡(π(x−[x2]))dx\int_0^5 \cos\left(\pi\left(x - \left[\frac{x}{2}\right]\right)\right)dx, where [t][t] denotes greatest integer less than or equal to…MediumSingle correct

2021 9 questions

  1. 3 Aug 2021, Shift 2 · Q72The integral ∫dx2x+5−2x+3\int \frac{dx}{\sqrt{2x+5}-\sqrt{2x+3}} is equal to (where C is a constant of integration)MediumSingle correct
  2. 3 Aug 2021, Shift 2 · Q84If [x][x] denotes the greatest integer ≤x\le x, then the value of ∫−24∣[x]−x∣dx\int_{-2}^{4}\left|[x] - x\right|dx is ‾\underline{\hspace{2em}}.MediumNumerical value
  3. 25 Jul 2021, Shift 1 · Q78The value of the definite integral ∫π/245π/24dx1+tan⁡2x3\int_{\pi/24}^{5\pi/24}\frac{dx}{1+\sqrt[3]{\tan2x}} is :MediumSingle correct
  4. 16 Mar 2021, Shift 2 · Q72Let P(x)=x2+bx+cP(x)=x^2+bx+c be a quadratic polynomial with real coefficients such that ∫01P(x) dx=1\int_0^1P(x)\,dx=1 and P(x)P(x) leaves remainder 5 when it…EasySingle correct
  5. 16 Mar 2021, Shift 2 · Q74Consider the integral I=∫010[x] e[x]ex−1 dxI=\int_0^{10}\frac{[x]\,e^{[x]}}{e^{x-1}}\,dx, where [x][x] denotes the greatest integer less than or equal to xx.…MediumSingle correct
  6. 16 Mar 2021, Shift 2 · Q81For real numbers α,β,γ\alpha,\beta,\gamma and δ\delta, if…HardNumerical value
  7. 18 Mar 2021, Shift 1 · Q73The integral ∫(2x−1)cos⁡(2x−1)2+54x2−4x+6 dx\int\dfrac{(2x-1)\cos\sqrt{(2x-1)^2+5}}{\sqrt{4x^2-4x+6}}\,dx is equal to : (where c is a constant of integration)MediumSingle correct
  8. 18 Mar 2021, Shift 1 · Q82Let f(x)f(x) and g(x)g(x) be two functions satisfying f(x2)+g(4−x)=4x3f(x^2)+g(4-x)=4x^3 and g(4−x)+g(x)=0g(4-x)+g(x)=0, then the value of ∫−44f(x2) dx\int_{-4}^{4}f(x^2)\,dx is…HardNumerical value
  9. 18 Mar 2021, Shift 1 · Q83If f(x)=∫5x8+7x6(x2+1+2x7)2 dxf(x)=\int\dfrac{5x^8+7x^6}{(x^2+1+2x^7)^2}\,dx, (x≥0)(x\ge0), f(0)=0f(0)=0 and f(1)=1Kf(1)=\dfrac{1}{K}, then the value of K is __________.HardNumerical value

2020 4 questions

  1. 8 Jan 2020, Shift 1 · Q60If ∫cos⁡x dxsin⁡3x (1+sin⁡6x)2/3=f(x) (1+sin⁡6x)1/λ+c\int\dfrac{\cos x\,dx}{\sin^3 x\,(1+\sin^6 x)^{2/3}}=f(x)\,(1+\sin^6 x)^{1/\lambda}+c where c is a constant of integration, then…HardSingle correct
  2. 9 Jan 2020, Shift 2 · Q62If ∫dθcos⁡2θ(tan⁡2θ+sec⁡2θ)=λtan⁡θ+2log⁡e∣f(θ)∣+C\int\frac{d\theta}{\cos^2\theta(\tan2\theta+\sec2\theta)}=\lambda\tan\theta+2\log_e|f(\theta)|+C where CC is a constant of…MediumSingle correct
  3. 2 Sep 2020, Shift 1 · Q73The integral ∫02∣∣x−1∣−x∣dx\int_{0}^{2}\left||x-1|-x\right|dx is equal to ___________.EasyNumerical value
  4. 6 Sep 2020, Shift 2 · Q61The integral ∫12ex⋅xx(2+log⁡ex) dx\int_1^2 e^x\cdot x^x(2+\log_e x)\,dx equals :HardSingle correct

2017 2 questions

  1. 2 Apr 2017 (offline) · Q74Let In=∫tan⁡nx dx, (n>1)I_n=\int\tan^n x\,dx,\ (n>1). If I4+I6=atan⁡5x+bx5+CI_4+I_6=a\tan^5x+bx^5+C, where C is a constant of integration, then the ordered pair (a, b) is…MediumSingle correct
  2. 2 Apr 2017 (offline) · Q75The integral ∫π/43π/4dx1+cos⁡x\int_{\pi/4}^{3\pi/4}\frac{dx}{1+\cos x} is equal to :EasySingle correct

Pattern: Asked most years.