Integral Calculus & Area · JEE Main previous year questions

2025 30 questions

1. 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

2. 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

3. 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

4. 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

5. Q152025 · 22 Jan 2025 · Shift 1

   Let for $f(x)=7{\tan }^{8}x+7{\tan }^{6}x-3{\tan }^{4}x-3{\tan }^{2}x$, ${\mathrm{I}}_1={\int }_0^{\pi /4}f(x)dx$ and ${\mathrm{I}}_2={\int }_0^{\pi /4}xf(x)dx$. Then $7{\mathrm{I}}_1+12{\mathrm{I}}_2$ is equal to :

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