Integral Calculus & Area · JEE Main previous year questions

2025 30 questions

1. Q102025 · 4 Apr 2025 · Shift 1

   Let $f:[0,\infty )\to \mathbb{R}$ be a differentiable function such that $f(x)=1-2x+{\int }_0^x{e}^{x-t}f(t)dt$ for all $x\in [0,\infty )$. Then the area of the region bounded by $y=f(x)$ and the coordinate axes is

   1. A.$\sqrt{5}$
   2. B.$2$
   3. C.$\sqrt{2}$
   4. D.$\frac{1}{2}$
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   Answer: D

2. Q242025 · 4 Apr 2025 · Shift 1· Numerical answer

   If the area of the region $\{(x,y):|x-5|\le y\le 4\sqrt{x}\}$ is $A$, then $3A$ is equal to _________.

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   Answer: 368

3. Q172025 · 4 Apr 2025 · Shift 2

   Let $f(x)+2f\left(\frac{1}{x}\right)=x^2+5$ and $2g(x)-3g\left(\frac{1}{x}\right)=x,x>0$. If $\alpha ={\int }_1^{2}f(x)dx$, and $\beta ={\int }_1^{2}g(x)dx$, then the value of $9\alpha +\beta$ is :

   1. A.0
   2. B.10
   3. C.1
   4. D.11
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   Answer: D

4. Q212025 · 4 Apr 2025 · Shift 2· Numerical answer

   If $\int \frac{{\left(\sqrt{1+x^2}+x\right)}^{10}}{{\left(\sqrt{1+x^2}-x\right)}^9}dx=\frac{1}{\mathrm{m}}\left({\left(\sqrt{1+x^2}+x\right)}^{\mathrm{n}}\left(\mathrm{n}\sqrt{1+x^2}-x\right)\right)+\mathrm{C}$ where C is the constant of integration and $\mathrm{m},\mathrm{n}\in \mathbf{N}$, then $\mathrm{m}+\mathrm{n}$ is equal to ____ .

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   Answer: 379

5. Q72025 · 3 Apr 2025 · Shift 1

   Let $f(x)=\int x^3\sqrt{3-x^2}dx$. If $5f(\sqrt{2})=-4$, then $f(1)$ is equal to

   1. A.$-\frac{6\sqrt{2}}{5}$
   2. B.$-\frac{8\sqrt{2}}{5}$
   3. C.$-\frac{2\sqrt{2}}{5}$
   4. D.$-\frac{4\sqrt{2}}{5}$
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   Answer: A

6. Q242025 · 3 Apr 2025 · Shift 1· Numerical answer

   The area of the region bounded by the curve $y=\max \{|x|,x|x-2|\}$, the $x$-axis and the lines $x=-2$ and $x=4$ is equal to____

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   Answer: 12

7. Q12025 · 3 Apr 2025 · Shift 2

   The area of the region $\{(x,y):|x-y|\le y\le 4\sqrt{x}\}$ is

   1. A.$\frac{512}{3}$
   2. B.$\frac{2048}{3}$
   3. C.$512$
   4. D.$\frac{1024}{3}$
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   Answer: D

8. Q132025 · 3 Apr 2025 · Shift 2

   The integral ${\int }_0^{\pi }\frac{8xdx}{4{\cos }^{2}x+{\sin }^{2}x}$ is equal to

   1. A.$2{\pi }^2$
   2. B.$4{\pi }^2$
   3. C.${\pi }^2$
   4. D.$\frac{3{\pi }^2}{2}$
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   Answer: A

9. Q232025 · 2 Apr 2025 · Shift 1· Numerical answer

   If the area of the region $\{(x,y):|4-x^2|\le y\le x^2,y\le 4,x\ge 0\}$ is $\left(\frac{80\sqrt{2}}{\alpha }-\beta \right),\alpha ,\beta \in \mathbf{N}$, then $\alpha +\beta$ is equal to _________ .

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   Answer: 22

10. Q22025 · 2 Apr 2025 · Shift 2

    Let $f:[1,\infty )\to [2,\infty )$ be a differentiable function. If $10{\int }_1^{x}f(\mathrm{t})\mathrm{dt}=5xf(x)-x^5-9$ for all $x\ge 1$, then the value of $f(3)$ is :

    1. A.22
    2. B.26
    3. C.32
    4. D.18
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    Answer: C

11. Q152025 · 2 Apr 2025 · Shift 2

    Let $(a,b)$ be the point of intersection of the curve $x^2=2y$ and the straight line $y-2x-6=0$ in the second quadrant. Then the integral $\mathrm{I}={\int }_a^b\frac{9x^2}{1+5^x}dx$ is equal to :

    1. A.27
    2. B.18
    3. C.24
    4. D.21
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    Answer: C

12. Q182025 · 2 Apr 2025 · Shift 2

    $4{\int }_0^1\left(\frac{1}{\sqrt{3+x^2}+\sqrt{1+x^2}}\right)dx-3{\log }_e(\sqrt{3})$ is equal to :

    1. A.$2-\sqrt{2}-{\log }_e(1+\sqrt{2})$
    2. B.$2+\sqrt{2}+{\log }_e(1+\sqrt{2})$
    3. C.$2+\sqrt{2}-{\log }_e(1+\sqrt{2})$
    4. D.$2-\sqrt{2}+{\log }_e(1+\sqrt{2})$
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    Answer: A

13. Q162025 · 29 Jan 2025 · Shift 2

    Let $f(x)=\underset{0}{\overset{x}{\int }}t\left(t^2-9t+20\right)\mathrm{dt}$, $1\le x\le 5$. If the range of $f$ is $[\alpha ,\beta ]$, then $4(\alpha +\beta )$ equals :

    1. A.$253$
    2. B.$157$
    3. C.$154$
    4. D.$125$
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    Answer: B

14. Q172025 · 29 Jan 2025 · Shift 2

    Let the area enclosed between the curves $|y|=1-x^2$ and $x^2+y^2=1$ be $\alpha$. If $9\alpha =\beta \pi +\gamma$; $\beta ,\gamma$ are integers, then the value of $|\beta -\gamma |$ equals:

    1. A.$15$
    2. B.$18$
    3. C.$33$
    4. D.$27$
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    Answer: C

15. Q142025 · 28 Jan 2025 · Shift 1

    Let for some function $y=f(x)$, ${\int }_0^{x}tf(t)\mathrm{d}t=x^{2}f(x),x>0$ and $f(2)=3$. Then $f(6)$ is equal to

    1. A.1
    2. B.6
    3. C.2
    4. D.3
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    Answer: A

16. Q62025 · 24 Jan 2025 · Shift 1

    If $I(m,n)={\int }_0^1{x}^{m-1}(1-x{)}^{n-1}dx,m,n>0$, then $I(9,14)+I(10,13)$ is

    1. A.$I(9,1)$
    2. B.$I(1,13)$
    3. C.$I(19,27)$
    4. D.$I(9,13)$
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    Answer: D

17. Q92025 · 24 Jan 2025 · Shift 1

    Let $y=y(x)$ be the solution of the differential equation $\left(xy-5x^2\sqrt{1+x^2}\right)dx+\left(1+x^2\right)dy=0,y(0)=0$. Then $y(\sqrt{3})$ is equal to

    1. A.$\frac{5\sqrt{3}}{2}$
    2. B.$\sqrt{\frac{15}{2}}$
    3. C.$\sqrt{\frac{14}{3}}$
    4. D.$2\sqrt{2}$
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    Answer: A

18. Q142025 · 24 Jan 2025 · Shift 1

    Consider the region $R=\left\{(x,y):x\le y\le 9-\frac{11}{3}{x}^2,x\ge 0\right\}$. The area of the largest rectangle of sides parallel to the coordinate axes and inscribed in R , is:

    1. A.$\frac{821}{123}$
    2. B.$\frac{567}{121}$
    3. C.$\frac{730}{119}$
    4. D.$\frac{625}{111}$
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    Answer: B

19. Q222025 · 24 Jan 2025 · Shift 1· Numerical answer

    Let $f$ be a differentiable function such that $2(x+2{)}^{2}f(x)-3(x+2{)}^2=10{\int }_0^x(t+2)f(t)dt,x\ge 0$. Then $f(2)$ is equal to ____ .

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    Answer: 19

20. Q52025 · 24 Jan 2025 · Shift 2

    The area of the region enclosed by the curves $y={\mathrm{e}}^x$, $y=|{\mathrm{e}}^x-1|$ and $y$-axis is :

    1. A.$1+{\log }_{\mathrm{e}}2$
    2. B.${\log }_{\mathrm{e}}2$
    3. C.$1-{\log }_{\mathrm{e}}2$
    4. D.$2{\log }_{\mathrm{e}}2-1$
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    Answer: C

21. Q222025 · 24 Jan 2025 · Shift 2· Numerical answer

    If $\int \frac{2x^2+5x+9}{\sqrt{x^2+x+1}}dx=x\sqrt{x^2+x+1}+\alpha \sqrt{x^2+x+1}+\beta {\log }_e\left|x+\frac{1}{2}+\sqrt{x^2+x+1}\right|+C$, where $C$ is the constant of integration, then $\alpha +2\beta$ is equal to ________ .

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    Answer: 16

22. Q12025 · 23 Jan 2025 · Shift 2

    If $\mathrm{I}={\int }_0^{\frac{\pi }{2}}\frac{{\sin }^{\frac{3}{2}}x}{{\sin }^{\frac{3}{2}}x+{\cos }^{\frac{3}{2}}x}\mathrm{d}x$, then ${\int }_0^{2\mathrm{I}}\frac{x\sin x\cos x}{{\sin }^{4}x+{\cos }^{4}x}\mathrm{d}x$ equals :

    1. A.$\frac{{\pi }^2}{12}$
    2. B.$\frac{{\pi }^2}{4}$
    3. C.$\frac{{\pi }^2}{16}$
    4. D.$\frac{{\pi }^2}{8}$
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    Answer: C

23. Q82025 · 23 Jan 2025 · Shift 2

    If the area of the region $\{(x,y):-1\le x\le 1,0\le y\le a+{\mathrm{e}}^{|x|}-{\mathrm{e}}^{-x},a>0\}$ is $\frac{{\mathrm{e}}^2+8\mathrm{e}+1}{\mathrm{e}}$, then the value of $a$ is :

    1. A.7
    2. B.5
    3. C.6
    4. D.8
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    Answer: B

24. Q172025 · 23 Jan 2025 · Shift 2

    Let $\int x^3\sin x\mathrm{d}x=g(x)+C$, where $C$ is the constant of integration. If $8\left(g\left(\frac{\pi }{2}\right)+g^{\prime }\left(\frac{\pi }{2}\right)\right)=\alpha {\pi }^3+\beta {\pi }^2+\gamma ,\alpha ,\beta ,\gamma \in \mathbf{Z}$, then $\alpha +\beta -\gamma$ equals :

    1. A.47
    2. B.55
    3. C.62
    4. D.48
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    Answer: B

25. Q102025 · 22 Jan 2025 · Shift 1

    Let $x=x(y)$ be the solution of the differential equation $y^{2}dx+\left(x-\frac{1}{y}\right)dy=0$. If $x(1)=1$, then $x\left(\frac{1}{2}\right)$ is :

    1. A.$\frac{3}{2}+e$
    2. B.$\frac{1}{2}+e$
    3. C.$3+e$
    4. D.$3-e$
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    Answer: D