Integral Calculus & Area · JEE Main previous year questions

2026 46 questions

1. Q162026 · 8 Apr 2026 · Shift 2

   The value of the integral ${\int }_0^2\frac{\sqrt{x}(x^2+x+1)}{(\sqrt{x}+1)(\sqrt{x^4+x^2+1})}dx$ is equal to :

   1. A.$\frac{1}{3}{\log }_e(3-2\sqrt{2})$
   2. B.$\frac{2}{3}{\log }_e(4+\sqrt{2})$
   3. C.$\frac{2}{3}{\log }_e(3+2\sqrt{2})$
   4. D.$\frac{1}{3}{\log }_e(1+6\sqrt{2})$
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   Answer: C

2. Q172026 · 8 Apr 2026 · Shift 2

   Let $y=y(x)$ be the solution of the differential equation $x\sqrt{1-x^2}dy+\left(y\sqrt{1-x^2}-x{\cos }^{-1}x\right)dx=0$, $x\in (0,1)$, ${\lim }_{x\to 1^{-}}y(x)=1$. Then $y\left(\frac{1}{2}\right)$ equals :

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

3. Q232026 · 8 Apr 2026 · Shift 2· Numerical answer

   Let a line ${\mathrm{L}}_1$ pass through the origin and be perpendicular to the lines ${\mathrm{L}}_2:\vec{r}=(3+t)\hat{i}+(2t-1)\hat{j}+(2t+4)\hat{k}$ and ${\mathrm{L}}_3:\vec{r}=(3+2s)\hat{i}+(3+2s)\hat{j}+(2+s)\hat{k}$, $t,s\in \mathbf{R}$. If $(a,b,c)$, $a\in \mathbf{Z}$, is the point on ${\mathrm{L}}_3$ at a distance of $\sqrt{17}$ from the point of intersection of ${\mathrm{L}}_1$ and ${\mathrm{L}}_2$, then $(a+b+c{)}^2$ is equal to ____ .

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   Answer: 4

4. Q152026 · 6 Apr 2026 · Shift 1

   Let the image of the point $\mathrm{P}(1,6,a)$ in the line $\mathrm{L}:\frac{x}{1}=\frac{y-1}{2}=\frac{z-a+1}{b}$, $b>0$, be $\left(\frac{a}{3},0,a+c\right)$. If $\mathrm{S}(\alpha ,\beta ,\gamma )$, $a>0$, is the point on L such that the distance of S from the foot of perpendicular from the point P on L is $2\sqrt{14}$, then $\alpha +\beta +\gamma$ is equal to:

   1. A.19
   2. B.20
   3. C.21
   4. D.22
   Show answer

   Answer: C

5. Q182026 · 6 Apr 2026 · Shift 1

   The value of the integral ${\int }_{-\frac{\pi }{4}}^{\frac{\pi }{4}}\left(\frac{32{\cos }^{4}x}{1+e^{\sin x}}\right)dx$ is:

   1. A.$4\pi +2$
   2. B.$3\pi +8$
   3. C.$3\pi +4$
   4. D.$4\pi +3$
   Show answer

   Answer: B

6. Q192026 · 6 Apr 2026 · Shift 1

   The area of the region $\{(x,y):0\le y\le 6-x,y^2\ge 4x-3,x\ge 0\}$ is:

   1. A.8
   2. B.9
   3. C.12
   4. D.15
   Show answer

   Answer: B

7. Q182026 · 6 Apr 2026 · Shift 2

   Let $f:\mathbf{R}\to \mathbf{R}$ be such that $f(xy)=f(x)f(y)$, for all $x,y\in \mathbf{R}$ and $f(0)\ne 0$. Let $g:[1,\infty )\to \mathbf{R}$ be a differentiable function such that $$x^{2}g(x)={\int }_1^x(t^{2}f(t)-tg(t))dt.$$ Then $g(2)$ is equal to :

   1. A.$\frac{13}{8}$
   2. B.$\frac{11}{16}$
   3. C.$\frac{15}{32}$
   4. D.$\frac{17}{64}$
   Show answer

   Answer: C

8. Q202026 · 6 Apr 2026 · Shift 2

   The value of the integral ${\int }_{-1}^1\left(\frac{x^3+|x|+1}{x^2+2|x|+1}\right)dx$ is equal to :

   1. A.$3{\log }_{e}2$
   2. B.$2{\log }_{e}2$
   3. C.$5{\log }_{e}3$
   4. D.$3{\log }_{e}3$
   Show answer

   Answer: B

9. Q122026 · 5 Apr 2026 · Shift 1

   The sum of all the integral values of $p$ such that the equation $3{\sin }^{2}x+12\cos x-3=p$, $x\in \mathbb{R}$, has at least one solution, is:

   1. A.-54
   2. B.-60
   3. C.-75
   4. D.-84
   Show answer

   Answer: C

10. Q162026 · 5 Apr 2026 · Shift 1

    The area of the region $\mathrm{R}=\{(x,y):xy\le 27,1\le y\le x^2\}$ is equal to:

    1. A.$78{\log }_{e}3-\frac{52}{3}$
    2. B.$54{\log }_{e}3-\frac{52}{3}$
    3. C.$54{\log }_{e}3-\frac{26}{3}$
    4. D.$54{\log }_{e}3+\frac{26}{3}$
    Show answer

    Answer: B

11. Q182026 · 5 Apr 2026 · Shift 1

    The value of the integral ${\int }_0^{\infty }\frac{{\log }_e(x)}{x^2+4}dx$ is:

    1. A.$\frac{\pi {\log }_e(2)}{2}$
    2. B.$\frac{\pi {\log }_e(2)}{4}$
    3. C.$1+\pi {\log }_e(2)$
    4. D.$2+\pi {\log }_e(2)$
    Show answer

    Answer: B

12. Q202026 · 5 Apr 2026 · Shift 1

    Given below are two statements : Statement I : Sodium dichromate and potassium dichromate are classified as primary standards in titrimetric analysis. Statement II : Phenolphthalein is a weak base, therefore it dissociates in acidic medium. In the light of the above statements, choose the correct answer from the options given below

    1. A.Both Statement I and Statement II are ture
    2. B.Both Statement I and Statement II are false
    3. C.Statement I is true but Statement II is false
    4. D.Statement I is false but Statement II is true
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    Answer: C

13. Q182026 · 5 Apr 2026 · Shift 2

    Identify the incorrect statement about tertiary structure of proteins.

    1. A.They can be fibrous or globular in structure.
    2. B.The main forces that stabilize the structure are hydrogen bonding, disulphide links, van der Waals and electrostatic forces of attraction.
    3. C.The structure remains intact when exposed to pH changes.
    4. D.A linear polypeptide chain will convert to a secondary structure and then further folding of the secondary structure will convert to tertiary structure.
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    Answer: B

14. Q192026 · 5 Apr 2026 · Shift 2

    Let $f:[1,\infty )\to \mathbb{R}$ be a differentiable function defined as $f(x)={\int }_1^{x}f(t)\mathrm{d}t+(1-x)({\log }_{\mathrm{e}}x-1)+\mathrm{e}$. Then the value of $f(f(1))$ is :

    1. A.$(1+{\mathrm{e}}^{\mathrm{e}})$
    2. B.$(1+\mathrm{e})$
    3. C.$(1+\mathrm{e}+{\mathrm{e}}^{\mathrm{e}})$
    4. D.$1+2\mathrm{e}$
    Show answer

    Answer: A

15. Q162026 · 4 Apr 2026 · Shift 1

    If $y={\tan }^{-1}\left(\frac{3\cos x-4\sin x}{4\cos x+3\sin x}\right)+2{\tan }^{-1}\left(\frac{x}{1+\sqrt{1-x^2}}\right)$, then $\frac{dy}{dx}$ at $x=\frac{\sqrt{3}}{2}$ is equal to:

    1. A.3
    2. B.$-1$
    3. C.1
    4. D.2
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    Answer: C

16. Q172026 · 4 Apr 2026 · Shift 1

    Let $f$ be a real polynomial of degree $n$ such that $f(x)=f^{\prime }(x)f^{\prime \prime }(x)$, for all $x\in \mathbb{R}$. If $f(0)=0$, then $36\left(f^{\prime }(2)+f^{\prime \prime }(2)+{\int }_0^{2}f(x)dx\right)$ is equal to:

    1. A.42
    2. B.46
    3. C.56
    4. D.66
    Show answer

    Answer: C

17. Q182026 · 4 Apr 2026 · Shift 1

    The area of the region $\{(x,y):y\le \pi -|x|,y\le |x\sin x|,y\ge 0\}$ is:

    1. A.$1+\frac{{\pi }^2}{8}$
    2. B.$2+\frac{{\pi }^2}{4}$
    3. C.$\frac{{\pi }^2}{8}-1$
    4. D.$4+\frac{{\pi }^2}{2}$
    Show answer

    Answer: B

18. Q182026 · 4 Apr 2026 · Shift 2

    The area of the region bounded by the curves $x+3y^2=0$ and $x+4y^2=1$ is equal to:

    1. A.$\frac{1}{3}$
    2. B.$\frac{2}{3}$
    3. C.$\frac{4}{3}$
    4. D.$\frac{5}{3}$
    Show answer

    Answer: C

19. Q202026 · 4 Apr 2026 · Shift 2

    The integral ${\int }_0^1{\cot }^{-1}\left(1+x+x^2\right)\mathrm{d}x$ is equal to:

    1. A.$2{\tan }^{-1}2+\frac{1}{2}{\log }_e\left(\frac{5}{4}\right)+\frac{\pi }{2}$
    2. B.$2{\tan }^{-1}2+\frac{1}{2}{\log }_e\left(\frac{5}{4}\right)-\frac{\pi }{2}$
    3. C.$2{\tan }^{-1}2-\frac{1}{2}{\log }_e\left(\frac{5}{4}\right)+\frac{\pi }{2}$
    4. D.$2{\tan }^{-1}2-\frac{1}{2}{\log }_e\left(\frac{5}{4}\right)-\frac{\pi }{2}$
    Show answer

    Answer: D

20. Q192026 · 2 Apr 2026 · Shift 1

    Let $[\cdot ]$ denote the greatest integer function. Then the value of ${\int }_0^3\left[\frac{{\mathrm{e}}^x+{\mathrm{e}}^{-x}}{[x]!}\right]\mathrm{d}x$ is :

    1. A.${\mathrm{e}}^2+{\mathrm{e}}^3-\frac{1}{{\mathrm{e}}^2}-\frac{1}{{\mathrm{e}}^3}$
    2. B.$\frac{1}{2}\left({\mathrm{e}}^2+{\mathrm{e}}^3-\frac{1}{{\mathrm{e}}^2}-\frac{1}{{\mathrm{e}}^3}\right)$
    3. C.${\mathrm{e}}^2+{\mathrm{e}}^3-\frac{1}{2{\mathrm{e}}^2}-\frac{1}{2{\mathrm{e}}^3}$
    4. D.$\frac{1}{2}\left({\mathrm{e}}^2+{\mathrm{e}}^3\right)-\frac{1}{{\mathrm{e}}^2}-\frac{1}{{\mathrm{e}}^3}$
    Show answer

    Answer: B

21. Q252026 · 2 Apr 2026 · Shift 1· Numerical answer

    If $\alpha ={\int }_0^{2\sqrt{3}}{\log }_2\left(x^2+4\right)\mathrm{d}x+{\int }_2^4\sqrt{2^x-4}\mathrm{d}x$, then ${\alpha }^2$ is equal to ____.

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    Answer: 192

22. Q172026 · 2 Apr 2026 · Shift 2

    The value of ${\int }_0^{20\pi }({\sin }^{4}x+{\cos }^{4}x)\mathrm{d}x$ is equal to :

    1. A.$\frac{15\pi }{2}$
    2. B.$25\pi$
    3. C.$15\pi$
    4. D.$\frac{25\pi }{2}$
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    Answer: C

23. Q192026 · 2 Apr 2026 · Shift 2

    Let $f(x)=\int \left(\frac{16x+24}{x^2+2x-15}\right)\mathrm{d}x$. If $f(4)=14{\log }_e(3)$ and $f(7)={\log }_e(2^{\alpha }\cdot 3^{\beta })$, $\alpha ,\beta \in \mathbb{N}$, then $\alpha +\beta$ is equal to :

    1. A.31
    2. B.37
    3. C.39
    4. D.41
    Show answer

    Answer: C

24. Q202026 · 2 Apr 2026 · Shift 2

    Let $x=x(y)$ be the solution of the differential equation $2y^2\frac{\mathrm{d}x}{\mathrm{d}y}-2xy+x^2=0$, $y>1$, $x(\mathrm{e})=\mathrm{e}$. Then $x({\mathrm{e}}^2)$ is equal to :

    1. A.$\frac{3}{2}{\mathrm{e}}^2$
    2. B.$\frac{2}{3}{\mathrm{e}}^2$
    3. C.${\mathrm{e}}^2$
    4. D.$2{\mathrm{e}}^2$
    Show answer

    Answer: B

25. Q12026 · 28 Jan 2026 · Shift 1

    The area of the region $\mathrm{R}=\left\{(x,y):xy\le 8,1\le y\le x^2,x\ge 0\right\}$ is

    1. A.$\frac{2}{3}(20{\log }_e(2)+9)$
    2. B.$\frac{1}{3}(40{\log }_e(2)+27)$
    3. C.$\frac{1}{3}(49{\log }_e(2)-15)$
    4. D.$\frac{2}{3}(24{\log }_e(2)-7)$
    Show answer

    Answer: D