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Definite Integration

JEE Mains Maths. Try each one first, then open its answer.

Top 10 JEE Mains PYQs to practise first

The questions and kinds of question asked most often come first. Each one says how often and in which papers.

  1. Question 1

    This type asked 11 times: 2021 Paper 16, 2021 Paper 19, 2021 Paper 25, 14 Jun 2022, 27 Jun 2022, 29 Jan 2024, 1 Feb 2024, 4 Apr 2024, 28 Jan 2025, 2 Apr 2025, 6 Apr 2026 S1. This one: 6 Apr 2026 S1 See all 11 of this type

    68
    The 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 :

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  2. Question 2

    This type asked 7 times: 26 Jun 2022, 31 Jan 2024, 1 Feb 2024, 5 Apr 2024, 23 Jan 2025, 3 Apr 2025, 7 Apr 2025. This one: 7 Apr 2025 See all 7 of this type

    60
    The integral ∫0π(x+3)sin⁡x1+3cos⁡2xdx\int_{0}^{\pi} \frac{(x+ 3)\sin x}{1 + 3\cos^{2}x}dx is equal to :

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  3. Question 3

    This type asked 6 times: 2021 Paper 7, 2021 Paper 14, 2021 Paper 20, 24 Jan 2023, 23 Jan 2025, 23 Jan 2026 S1. This one: 23 Jan 2026 S1 See all 6 of this type

    60
    The value of the integral ∫π245π24dx1+tan⁡2x3\int_{\frac{\pi}{24}}^{\frac{5\pi}{24}} \frac{dx}{1 +\sqrt[3]{\tan2x}} is :

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  4. Question 4

    This type asked 6 times: 2021 Paper 5, 14 Jun 2022, 26 Jul 2022, 30 Jan 2023, 30 Jan 2024, 9 Apr 2024. This one: 9 Apr 2024 See all 6 of this type

    84
    Let
    lim⁡n→∞(nn4+1−2n(n2+1)n4+1+nn4+16−8n(n2+4)n4+16\lim_{n \rightarrow \infty} \left( \frac{n}{\sqrt{n^{4}+ 1}}-\frac{2n}{\left( n^{2}+ 1 \right)\sqrt{n^{4}+ 1}}+\frac{n}{\sqrt{n^{4}+ 16}}-\frac{8n}{\left( n^{2}+ 4 \right)\sqrt{n^{4}+ 16}} \right.
    +……+nn4+n4−2n⋅n2(n2+n2)n4+n4+ \ldots\ldots +\frac{n}{\sqrt{n^{4}+n^{4}}}-\frac{2n \cdot n^{2}}{\left( n^{2}+n^{2} \right)\sqrt{n^{4}+n^{4}}} be πk\frac{\pi}{k}, using only the principal values of the inverse trigonometric functions. Then k2k^{2} is equal to............

  5. Question 5

    This type asked 6 times: 2021 Paper 6, 2021 Paper 8, 2021 Paper 11, 2021 Paper 13, 2021 Paper 19, 2021 Paper 25. This one: 2021 Paper 25 See all 6 of this type

    63
    The value of ∑n=1100∫n−1nex−[x]dx\sum_{n= 1}^{100} \int_{n- 1}^{n} e^{x- \lbrack x\rbrack}dx, where [x]\lbrack x\rbrack is the greatest integer ≤x\leq x, is:

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  6. Question 6

    This type asked 5 times: 2021 Paper 7, 2021 Paper 12, 25 Jul 2022, 8 Apr 2023, 28 Jan 2026 S2. This one: 28 Jan 2026 S2 See all 5 of this type

    63
    Let [ • ] denote the greatest integer function. Then ∫−π2π2(12(3+[x])3+[sin⁡x]+[cos⁡x])dx\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left( \frac{12(3 + \lbrack x\rbrack)}{3 + \lbrack \sin x\rbrack + \lbrack \cos x\rbrack} \right)dx is equal to:

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  7. Question 7

    This type asked 5 times: 2021 Paper 24, 14 Jun 2022, 29 Jun 2022, 29 Jan 2023, 24 Jan 2026 S2. This one: 24 Jan 2026 S2 See all 5 of this type

    71
    If f(x)f(x) satisfies the relation f(x)=ex+∫01(y+xex)f(y)dyf(x) = e^{x} + \int_{0}^{1} \left( y + {xe}^{x} \right)f(y)dy, then e+f(0)e + f(0) is equal to ____\_\_\_\_ .

  8. Question 8

    This type asked 4 times: 2021 Paper 26, 29 Jul 2022, 29 Jan 2023, 5 Apr 2026 S1. This one: 5 Apr 2026 S1 See all 4 of this type

    68
    The value of the integral ∫0∞log⁡e(x)x2+4dx\int_{0}^{\infty} \frac{\log_{e}(x)}{x^{2}+ 4}dx is:

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  9. Question 9

    This type asked 4 times: 10 Apr 2023, 24 Jan 2025, 28 Jan 2025, 2 Apr 2025. This one: 2 Apr 2025 See all 4 of this type

    127
    Let f:[1,∞)→[2,∞)f:\lbrack 1,\infty) \rightarrow \lbrack 2,\infty) be a differentiable function, If 10∫1xf(t)dt=5xf(x)−x5−910\int_{1}^{x} f(t)dt = 5xf(x) -x^{5}- 9 for all x≥1x \geq 1, then the value of f(3)f(3) is :

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  10. Question 10

    This type asked 4 times: 27 Jun 2022, 10 Apr 2023, 1 Feb 2024, 28 Jan 2025. This one: 28 Jan 2025 See all 4 of this type

    132
    Let ff be a real valued continuous function defined on the positive real axis such that g(x)=∫0xtf(t)dtg(x) =\int_{0}^{x} tf(t)dt. If g(x3)=x6+x7g\left( x^{3} \right)=x^{6}+x^{7}, then value of ∑r=115f(r3)\sum_{r = 1}^{15} f\left( r^{3} \right) is :

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