Question 1: Palette 1/76

JEE Mains 2026 MCQ

Let $ f(x)=\int \frac{d x}{x^{2 / 3}+2 x^{1 / 2}}, $ be such that $f(0) = -26 + 24 \log_e(2)$. If $f(1) = a + b \log_e(3)$, where $a, b \in \mathbb{Z}$, then $a + b$ is equal to :

A

$-5$

B

$-11$

C

$-18$

D

$-26$

Question 2: Palette 2/76

JEE Mains 2026 MCQ

If $\int\left(\frac{1-5 \cos ^2 x}{\sin ^5 x \cos ^2 x}\right) d x=f(x)+\mathrm{C}$, where C is the constant of integration, then $f\left(\frac{\pi}{6}\right)-f\left(\frac{\pi}{4}\right)$ is equal to

A

$\frac{1}{\sqrt{3}}(26-\sqrt{3})$

B

$\frac{4}{\sqrt{3}}(8-\sqrt{6})$

C

$\frac{1}{\sqrt{3}}(26+\sqrt{3})$

D

$\frac{2}{\sqrt{3}}(4+\sqrt{6})$

Question 3: Palette 3/76

JEE Mains 2026 MCQ

Let $f(t)=\int\left(\frac{1-\sin \left(\log _e t\right)}{1-\cos \left(\log _e t\right)}\right) d t, t>1$.

If $f\left(e^{\pi / 2}\right)=-e^{\pi / 2}$ and $f\left(e^{\pi / 4}\right)=\alpha e^{\pi / 4}$, then $\alpha$ equals

A

$1+\sqrt{2}$

B

$-1-2 \sqrt{2}$

C

$-1-\sqrt{2}$

D

$-1+\sqrt{2}$

Question 4: Palette 4/76

JEE Mains 2026 MCQ

Let $\mathrm{I}(x)=\int \frac{3 d x}{(4 x+6)\left(\sqrt{4 x^2+8 x+3}\right)}$ and $\mathrm{I}(0)=\frac{\sqrt{3}}{4}+20$. If

$\mathrm{I}\left(\frac{1}{2}\right)=\frac{a \sqrt{2}}{b}+\mathrm{c}$, where $a, b, \mathrm{c} \in \mathrm{N}, \operatorname{gcd}(a, b)=1$, then $a+b+c$ is equal to :

A

30

B

29

C

28

D

31

Question 5: Palette 5/76

JEE Mains 2026 MCQ

Let $f(x)=\int \frac{\left(2-x^2\right) \cdot \mathrm{e}^x}{(\sqrt{1+x})(1-x)^{3 / 2}} \mathrm{~d} x$. If $f(0)=0$, then $f\left(\frac{1}{2}\right)$ is equal to:

A

$\sqrt{2 \mathrm{e}}-1$

B

$\sqrt{2 \mathrm{e}}+1$

C

$\sqrt{3 \mathrm{e}}-1$

D

$\sqrt{3 \mathrm{e}}+1$

Question 6: Palette 6/76

JEE Mains 2026 MCQ

Let $f(x) = \int \left( \frac{16x + 24}{x^2 + 2x - 15} \right) dx$. 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 :

A

31

B

37

C

39

D

41

Question 7: Palette 7/76

JEE Mains 2025 MCQ

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

A

$-\frac{6 \sqrt{2}}{5}$

B

$-\frac{8 \sqrt{2}}{5}$

C

$-\frac{2 \sqrt{2}}{5}$

D

$-\frac{4 \sqrt{2}}{5}$

Question 8: Palette 8/76

JEE Mains 2025 MCQ

If $f(x)=\int \frac{1}{x^{1 / 4}\left(1+x^{1 / 4}\right)} \mathrm{d} x, f(0)=-6$, then $f(1)$ is equal to :

A

$4\left(\log _{\mathrm{e}} 2-2\right)$

B

$\log _{e^2} 2+2$

C

$2-\log \mathrm{e}^2$

D

$4\left(\log _e 2+2\right)$

Question 9: Palette 9/76

JEE Mains 2025 MCQ

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

A

47

B

55

C

62

D

48

Question 10: Palette 10/76

JEE Mains 2025 MCQ

Let $\mathrm{I}(x)=\int \frac{d x}{(x-11)^{\frac{11}{13}}(x+15)^{\frac{15}{13}}}$. If $\mathrm{I}(37)-\mathrm{I}(24)=\frac{1}{4}\left(\frac{1}{\mathrm{~b}^{\frac{1}{13}}}-\frac{1}{\mathrm{c}^{\frac{1}{13}}}\right), \mathrm{b}, \mathrm{c} \in \mathcal{N}$, then $3(\mathrm{~b}+\mathrm{c})$ is equal to

A

39

B

22

C

40

D

26

Question 11: Palette 11/76

JEE Mains 2025 MCQ

If $\int \mathrm{e}^x\left(\frac{x \sin ^{-1} x}{\sqrt{1-x^2}}+\frac{\sin ^{-1} x}{\left(1-x^2\right)^{3 / 2}}+\frac{x}{1-x^2}\right) \mathrm{d} x=\mathrm{g}(x)+\mathrm{C}$, where C is the constant of integration, then $g\left(\frac{1}{2}\right)$ equals :

A

$\frac{\pi}{6} \sqrt{\frac{\mathrm{e}}{3}}$

B

$\frac{\pi}{6} \sqrt{\frac{\mathrm{e}}{2}}$

C

$\frac{\pi}{4} \sqrt{\frac{\mathrm{e}}{3}}$

D

$\frac{\pi}{4} \sqrt{\frac{\mathrm{e}}{2}}$

Question 12: Palette 12/76

JEE Mains 2024 MCQ

Let $\int \frac{2-\tan x}{3+\tan x} \mathrm{~d} x=\frac{1}{2}\left(\alpha x+\log _e|\beta \sin x+\gamma \cos x|\right)+C$, where $C$ is the constant of integration. Then $\alpha+\frac{\gamma}{\beta}$ is equal to :

A

3

B

7

C

1

D

4

Question 13: Palette 13/76

JEE Mains 2024 MCQ

Let $I(x)=\int \frac{6}{\sin ^2 x(1-\cot x)^2} d x$. If $I(0)=3$, then $I\left(\frac{\pi}{12}\right)$ is equal to

A

$\sqrt3$

B

$2\sqrt3$

C

$6\sqrt3$

D

$3\sqrt3$

Question 14: Palette 14/76

JEE Mains 2024 MCQ

If $\int \frac{1}{\mathrm{a}^2 \sin ^2 x+\mathrm{b}^2 \cos ^2 x} \mathrm{~d} x=\frac{1}{12} \tan ^{-1}(3 \tan x)+$ constant, then the maximum value of $\mathrm{a} \sin x+\mathrm{b} \cos x$, is :

A

$\sqrt{41}$

B

$\sqrt{39}$

C

$\sqrt{40}$

D

$\sqrt{42}$

Question 15: Palette 15/76

JEE Mains 2024 MCQ

If $\int \frac{\sin ^{\frac{3}{2}} x+\cos ^{\frac{3}{2}} x}{\sqrt{\sin ^3 x \cos ^3 x \sin (x-\theta)}} d x=A \sqrt{\cos \theta \tan x-\sin \theta}+B \sqrt{\cos \theta-\sin \theta \cot x}+C$, where $C$ is the integration constant, then $A B$ is equal to

A

$2 \sec \theta$

B

$8 \operatorname{cosec}(2 \theta)$

C

$4 \operatorname{cosec}(2 \theta)$

D

$4 \sec \theta$

Question 16: Palette 16/76

JEE Mains 2024 MCQ

For $x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, if $y(x)=\int \frac{\operatorname{cosec} x+\sin x}{\operatorname{cosec} x \sec x+\tan x \sin ^2 x} d x$, and $\lim _\limits{x \rightarrow\left(\frac{\pi}{2}\right)^{-}} y(x)=0$ then $y\left(\frac{\pi}{4}\right)$ is equal to

A

$-\frac{1}{\sqrt{2}} \tan ^{-1}\left(\frac{1}{\sqrt{2}}\right)$

B

$\tan ^{-1}\left(\frac{1}{\sqrt{2}}\right)$

C

$\frac{1}{2} \tan ^{-1}\left(\frac{1}{\sqrt{2}}\right)$

D

$\frac{1}{\sqrt{2}} \tan ^{-1}\left(-\frac{1}{2}\right)$

Question 17: Palette 17/76

JEE Mains 2024 MCQ

$\text { The integral } \int \frac{\left(x^8-x^2\right) \mathrm{d} x}{\left(x^{12}+3 x^6+1\right) \tan ^{-1}\left(x^3+\frac{1}{x^3}\right)} \text { is equal to : }$

A

$\log _{\mathrm{e}}\left(\left|\tan ^{-1}\left(x^3+\frac{1}{x^3}\right)\right|\right)^{1 / 3}+\mathrm{C}$

B

$\log _{\mathrm{e}}\left(\left|\tan ^{-1}\left(x^3+\frac{1}{x^3}\right)\right|\right)+\mathrm{C}$

C

$\log _{\mathrm{e}}\left(\left|\tan ^{-1}\left(x^3+\frac{1}{x^3}\right)\right|\right)^{1 / 2}+\mathrm{C}$

D

$\log _{\mathrm{e}}\left(\left|\tan ^{-1}\left(x^3+\frac{1}{x^3}\right)\right|\right)^3+\mathrm{C}$

Question 18: Palette 18/76

JEE Mains 2023 MCQ

For $\alpha, \beta, \gamma, \delta \in \mathbb{N}$, if $\int\left(\left(\frac{x}{e}\right)^{2 x}+\left(\frac{e}{x}\right)^{2 x}\right) \log _{e} x d x=\frac{1}{\alpha}\left(\frac{x}{e}\right)^{\beta x}-\frac{1}{\gamma}\left(\frac{e}{x}\right)^{\delta x}+C$ , where $e=\sum_\limits{n=0}^{\infty} \frac{1}{n !}$ and $\mathrm{C}$ is constant of integration, then $\alpha+2 \beta+3 \gamma-4 \delta$ is equal to :

A

$-8$

B

$-4$

C

1

D

4

Question 19: Palette 19/76

JEE Mains 2023 MCQ

If $I(x) = \int {{e^{{{\sin }^2}x}}(\cos x\sin 2x - \sin x)dx} $ and $I(0) = 1$, then $I\left( {{\pi \over 3}} \right)$ is equal to :

A

$ - {e^{{3 \over 4}}}$

B

$ - {1 \over 2}{e^{{3 \over 4}}}$

C

${e^{{3 \over 4}}}$

D

${1 \over 2}{e^{{3 \over 4}}}$

Question 20: Palette 20/76

JEE Mains 2023 MCQ

The integral $ \int\left[\left(\frac{x}{2}\right)^x+\left(\frac{2}{x}\right)^x\right] \ln \left(\frac{e x}{2}\right) d x $ is equal to :

A

$\left(\frac{x}{2}\right)^{x}+\left(\frac{2}{x}\right)^{x}+C$

B

$\left(\frac{x}{2}\right)^{x}-\left(\frac{2}{x}\right)^{x}+C$

C

$\left(\frac{x}{2}\right)^{x} \log _{2}\left(\frac{2}{x}\right)+C$

D

None

Question 21: Palette 21/76

JEE Mains 2023 MCQ

Let $I(x)=\int \frac{(x+1)}{x\left(1+x e^{x}\right)^{2}} d x, x > 0$. If $\lim_\limits{x \rightarrow \infty} I(x)=0$, then $I(1)$ is equal to :

A

$\frac{e+1}{e+2}-\log _{e}(e+1)$

B

$\frac{e+1}{e+2}+\log _{e}(e+1)$

C

$\frac{e+2}{e+1}-\log _{e}(e+1)$

D

$\frac{e+2}{e+1}+\log _{e}(e+1)$

Question 22: Palette 22/76

JEE Mains 2023 MCQ

Let $I(x)=\int \frac{x^{2}\left(x \sec ^{2} x+\tan x\right)}{(x \tan x+1)^{2}} d x$. If $I(0)=0$, then $I\left(\frac{\pi}{4}\right)$ is equal to :

A

$\log _{e} \frac{(\pi+4)^{2}}{32}-\frac{\pi^{2}}{4(\pi+4)}$

B

$\log _{e} \frac{(\pi+4)^{2}}{16}-\frac{\pi^{2}}{4(\pi+4)}$

C

$\log _{e} \frac{(\pi+4)^{2}}{16}+\frac{\pi^{2}}{4(\pi+4)}$

D

$\log _{e} \frac{(\pi+4)^{2}}{32}+\frac{\pi^{2}}{4(\pi+4)}$

Question 23: Palette 23/76

JEE Mains 2023 MCQ

Let $f(x) = \int {{{2x} \over {({x^2} + 1)({x^2} + 3)}}dx} $. If $f(3) = {1 \over 2}({\log _e}5 - {\log _e}6)$, then $f(4)$ is equal to

A

${\log _e}19 - {\log _e}20$

B

${\log _e}17 - {\log _e}18$

C

${1 \over 2}({\log _e}19 - {\log _e}17)$

D

${1 \over 2}({\log _e}17 - {\log _e}19)$

Question 24: Palette 24/76

JEE Mains 2022 MCQ

For $I(x)=\int \frac{\sec ^{2} x-2022}{\sin ^{2022} x} d x$, if $I\left(\frac{\pi}{4}\right)=2^{1011}$, then

A

$3^{1010} I\left(\frac{\pi}{3}\right)-I\left(\frac{\pi}{6}\right)=0$

B

$3^{1010} I\left(\frac{\pi}{6}\right)-I\left(\frac{\pi}{3}\right)=0$

C

$3^{1011} I\left(\frac{\pi}{3}\right)-I\left(\frac{\pi}{6}\right)=0$

D

$3^{1011} I\left(\frac{\pi}{6}\right)-I\left(\frac{\pi}{3}\right)=0$

Question 25: Palette 25/76

JEE Mains 2022 MCQ

$ \text { The integral } \int \frac{\left(1-\frac{1}{\sqrt{3}}\right)(\cos x-\sin x)}{\left(1+\frac{2}{\sqrt{3}} \sin 2 x\right)} d x \text { is equal to } $

A

$\frac{1}{2} \log _{e}\left|\frac{\tan \left(\frac{x}{2}+\frac{\pi}{12}\right)}{\tan \left(\frac{x}{2}+\frac{\pi}{6}\right)}\right|+C$

B

$\frac{1}{2} \log _{e}\left|\frac{\tan \left(\frac{x}{2}+\frac{\pi}{6}\right)}{\tan \left(\frac{x}{2}+\frac{\pi}{3}\right)}\right|+C$

C

$ \log _{e}\left|\frac{\tan \left(\frac{x}{2}+\frac{\pi}{6}\right)}{\tan \left(\frac{x}{2}+\frac{\pi}{12}\right)}\right|+C$

D

$\frac{1}{2} \log _{e}\left|\frac{\tan \left(\frac{x}{2}-\frac{\pi}{12}\right)}{\tan \left(\frac{x}{2}-\frac{\pi}{6}\right)}\right|+C $

Question 26: Palette 26/76

JEE Mains 2022 MCQ

If $\int {{{({x^2} + 1){e^x}} \over {{{(x + 1)}^2}}}dx = f(x){e^x} + C} $, where C is a constant, then ${{{d^3}f} \over {d{x^3}}}$ at x = 1 is equal to :

A

$ - {3 \over 4}$

B

${3 \over 4}$

C

$ - {3 \over 2}$

D

${3 \over 2}$

Question 27: Palette 27/76

JEE Mains 2022 MCQ

If $\int {{1 \over x}\sqrt {{{1 - x} \over {1 + x}}} dx = g(x) + c} $, $g(1) = 0$, then $g\left( {{1 \over 2}} \right)$ is equal to :

A

${\log _e}\left( {{{\sqrt 3 - 1} \over {\sqrt 3 + 1}}} \right) + {\pi \over 3}$

B

${\log _e}\left( {{{\sqrt 3 + 1} \over {\sqrt 3 - 1}}} \right) + {\pi \over 3}$

C

${\log _e}\left( {{{\sqrt 3 + 1} \over {\sqrt 3 - 1}}} \right) - {\pi \over 3}$

D

${1 \over 2}{\log _e}\left( {{{\sqrt 3 - 1} \over {\sqrt 3 + 1}}} \right) - {\pi \over 6}$

Question 28: Palette 28/76

JEE Mains 2021 MCQ

The integral $\int {{1 \over {\root 4 \of {{{(x - 1)}^3}{{(x + 2)}^5}} }}} \,dx$ is equal to : (where C is a constant of integration)

A

${3 \over 4}{\left( {{{x + 2} \over {x - 1}}} \right)^{{1 \over 4}}} + C$

B

${3 \over 4}{\left( {{{x + 2} \over {x - 1}}} \right)^{{5 \over 4}}} + C$

C

${4 \over 3}{\left( {{{x - 1} \over {x + 2}}} \right)^{{1 \over 4}}} + C$

D

${4 \over 3}{\left( {{{x - 1} \over {x + 2}}} \right)^{{5 \over 4}}} + C$

Question 29: Palette 29/76

JEE Mains 2021 MCQ

The integral $\int {{{(2x - 1)\cos \sqrt {{{(2x - 1)}^2} + 5} } \over {\sqrt {4{x^2} - 4x + 6} }}} dx$ is equal to (where c is a constant of integration)

A

${1 \over 2}\sin \sqrt {{{(2x - 1)}^2} + 5} + c$

B

${1 \over 2}\cos \sqrt {{{(2x + 1)}^2} + 5} + c$

C

${1 \over 2}\cos \sqrt {{{(2x - 1)}^2} + 5} + c$

D

${1 \over 2}\sin \sqrt {{{(2x + 1)}^2} + 5} + c$

Question 30: Palette 30/76

JEE Mains 2021 MCQ

The integral $\int {{{{e^{3{{\log }_e}2x}} + 5{e^{2{{\log }_e}2x}}} \over {{e^{4{{\log }_e}x}} + 5{e^{3{{\log }_e}x}} - 7{e^{2{{\log }_e}x}}}}} dx$, x > 0, is equal to : (where c is a constant of integration)

A

${\log _e}\sqrt {{x^2} + 5x - 7} + c$

B

$4{\log _e}|{x^2} + 5x - 7| + c$

C

${1 \over 4}{\log _e}|{x^2} + 5x - 7| + c$

D

${\log _e}|{x^2} + 5x - 7| + c$

Question 31: Palette 31/76

JEE Mains 2021 MCQ

The value of the integral
$\int {{{\sin \theta .\sin 2\theta ({{\sin }^6}\theta + {{\sin }^4}\theta + {{\sin }^2}\theta )\sqrt {2{{\sin }^4}\theta + 3{{\sin }^2}\theta + 6} } \over {1 - \cos 2\theta }}} \,d\theta $ is :

A

${1 \over {18}}{\left[ {9 - 2{{\cos }^6}\theta - 3{{\cos }^4}\theta - 6{{\cos }^2}\theta } \right]^{{3 \over 2}}} + c$

B

${1 \over {18}}{\left[ {11 - 18{{\sin }^2}\theta + 9{{\sin }^4}\theta - 2{{\sin }^6}\theta } \right]^{{3 \over 2}}} + c$

C

${1 \over {18}}{\left[ {11 - 18{{\cos }^2}\theta + 9{{\cos }^4}\theta - 2{{\cos }^6}\theta } \right]^{{3 \over 2}}} + c$

D

${1 \over {18}}{\left[ {9 - 2{{\sin }^6}\theta - 3{{\sin }^4}\theta - 6{{\sin }^2}\theta } \right]^{{3 \over 2}}} + c$

Question 32: Palette 32/76

JEE Mains 2021 MCQ

If $\int {{{\cos x - \sin x} \over {\sqrt {8 - \sin 2x} }}} dx = a{\sin ^{ - 1}}\left( {{{\sin x + \cos x} \over b}} \right) + c$, where c is a constant of integration, then the ordered pair (a, b) is equal to :

A

(-1, 3)

B

(1, 3)

C

(1, -3)

D

(3, 1)

Question 33: Palette 33/76

JEE Mains 2020 MCQ

If
$\int {{{\cos \theta } \over {5 + 7\sin \theta - 2{{\cos }^2}\theta }}} d\theta $ = A${\log _e}\left| {B\left( \theta \right)} \right| + C$,

where C is a constant of integration, then ${{{B\left( \theta \right)} \over A}}$
can be :

A

${{2\sin \theta + 1} \over {5\left( {\sin \theta + 3} \right)}}$

B

${{2\sin \theta + 1} \over {\sin \theta + 3}}$

C

${{5\left( {2\sin \theta + 1} \right)} \over {\sin \theta + 3}}$

D

${{5\left( {\sin \theta + 3} \right)} \over {2\sin \theta + 1}}$

Question 34: Palette 34/76

JEE Mains 2020 MCQ

If
$\int {\left( {{e^{2x}} + 2{e^x} - {e^{ - x}} - 1} \right){e^{\left( {{e^x} + {e^{ - x}}} \right)}}dx} $ = $g\left( x \right){e^{\left( {{e^x} + {e^{ - x}}} \right)}} + c$

where c is a constant of integration, then g(0) is equal to :

A

1

B

2

C

e

D

e²

Question 35: Palette 35/76

JEE Mains 2020 MCQ

The integral $\int {{{\left( {{x \over {x\sin x + \cos x}}} \right)}^2}dx} $ is equal to
(where C is a constant of integration):

A

$\sec x - {{x\tan x} \over {x\sin x + \cos x}} + C$

B

$\sec x + {{x\tan x} \over {x\sin x + \cos x}} + C$

C

$\tan x - {{x\sec x} \over {x\sin x + \cos x}} + C$

D

$\tan x + {{x\sec x} \over {x\sin x + \cos x}} + C$

Question 36: Palette 36/76

JEE Mains 2020 MCQ

Let $f\left( x \right) = \int {{{\sqrt x } \over {{{\left( {1 + x} \right)}^2}}}dx\left( {x \ge 0} \right)} $. Then f(3) – f(1) is eqaul to :

A

$ - {\pi \over {12}} + {1 \over 2} + {{\sqrt 3 } \over 4}$

B

$ {\pi \over {12}} + {1 \over 2} - {{\sqrt 3 } \over 4}$

C

$ - {\pi \over 6} + {1 \over 2} + {{\sqrt 3 } \over 4}$

D

${\pi \over 6} + {1 \over 2} - {{\sqrt 3 } \over 4}$

Question 37: Palette 37/76

JEE Mains 2020 MCQ

If $\int {{{\sin }^{ - 1}}\left( {\sqrt {{x \over {1 + x}}} } \right)} dx$ = A(x)${\tan ^{ - 1}}\left( {\sqrt x } \right)$ + B(x) + C,
where C is a constant of integration, then the ordered pair (A(x), B(x)) can be :

A

(x + 1, -${\sqrt x }$)

B

(x + 1, ${\sqrt x }$)

C

(x - 1, -${\sqrt x }$)

D

(x - 1, ${\sqrt x }$)

Question 38: Palette 38/76

JEE Mains 2020 MCQ

If $\int {{{d\theta } \over {{{\cos }^2}\theta \left( {\tan 2\theta + \sec 2\theta } \right)}}} = \lambda \tan \theta + 2{\log _e}\left| {f\left( \theta \right)} \right| + C$

where C is a constant of integration, then the ordered pair ($\lambda $, ƒ($\theta $)) is equal to :

A

(–1, 1 – tan$\theta $)

B

(1, 1 + tan$\theta $)

C

(–1, 1 + tan$\theta $)

D

(1, 1 – tan$\theta $)

Question 39: Palette 39/76

JEE Mains 2020 MCQ

The integral $\int {{{dx} \over {{{(x + 4)}^{{8 \over 7}}}{{(x - 3)}^{{6 \over 7}}}}}} $ is equal to :
(where C is a constant of integration)

A

${1 \over 2}{\left( {{{x - 3} \over {x + 4}}} \right)^{{3 \over 7}}} + C$

B

${\left( {{{x - 3} \over {x + 4}}} \right)^{{1 \over 7}}} + C$

C

$ - {1 \over {13}}{\left( {{{x - 3} \over {x + 4}}} \right)^{{{13} \over 7}}} + C$

D

-${\left( {{{x - 3} \over {x + 4}}} \right)^{-{1 \over 7}}} + C$

Question 40: Palette 40/76

JEE Mains 2020 MCQ

If ƒ'(x) = tan^–1(secx + tanx), $ - {\pi \over 2} < x < {\pi \over 2}$,
and ƒ(0) = 0, then ƒ(1) is equal to :

A

${1 \over 4}$

B

${{\pi - 1} \over 4}$

C

${{\pi + 1} \over 4}$

D

${{\pi + 2} \over 4}$

Question 41: Palette 41/76

JEE Mains 2020 MCQ

If $\int {{{\cos xdx} \over {{{\sin }^3}x{{\left( {1 + {{\sin }^6}x} \right)}^{2/3}}}}} = f\left( x \right){\left( {1 + {{\sin }^6}x} \right)^{1/\lambda }} + c$

where c is a constant of integration, then $\lambda f\left( {{\pi \over 3}} \right)$ is equal to

A

${9 \over 8}$

B

2

C

-2

D

$-{9 \over 8}$

Question 42: Palette 42/76

JEE Mains 2019 MCQ

Let $a \in \left( {0,{\pi \over 2}} \right)$ be fixed. If the integral

$\int {{{\tan x + \tan \alpha } \over {\tan x - \tan \alpha }}} dx$ = A(x) cos 2$\alpha $ + B(x) sin 2$\alpha $ + C, where C is a

constant of integration, then the functions A(x) and B(x) are respectively :

A

$x - \alpha $ and ${\log _e}\left| {\cos \left( {x - \alpha } \right)} \right|$

B

$x + \alpha $ and ${\log _e}\left| {\sin \left( {x - \alpha } \right)} \right|$

C

$x + \alpha $ and ${\log _e}\left| {\sin \left( {x + \alpha } \right)} \right|$

D

$x - \alpha $ and ${\log _e}\left| {\sin \left( {x - \alpha } \right)} \right|$

Question 43: Palette 43/76

JEE Mains 2019 MCQ

The integral $\int {{{2{x^3} - 1} \over {{x^4} + x}}} dx$ is equal to :
(Here C is a constant of integration)

A

${\log _e}{{\left| {{x^3} + 1} \right|} \over {{x^2}}} + C$

B

${1 \over 2}{\log _e}{{\left| {{x^3} + 1} \right|} \over {{x^2}}} + C$

C

${\log _e}\left| {{{{x^3} + 1} \over x}} \right| + C$

D

${1 \over 2}{\log _e}{{{{\left( {{x^3} + 1} \right)}^2}} \over {\left| {{x^3}} \right|}} + C$

Question 44: Palette 44/76

JEE Mains 2019 MCQ

If $\int {{x^5}} {e^{ - {x^2}}}dx = g\left( x \right){e^{ - {x^2}}} + c$, where c is a constant of integration, then $g$(–1) is equal to :

A

1

B

- 1

C

$ - {5 \over 2}$

D

$ - {1 \over 2}$

Question 45: Palette 45/76

JEE Mains 2019 MCQ

If $\int {{{dx} \over {{{\left( {{x^2} - 2x + 10} \right)}^2}}}} = A\left( {{{\tan }^{ - 1}}\left( {{{x - 1} \over 3}} \right) + {{f\left( x \right)} \over {{x^2} - 2x + 10}}} \right) + C$

where C is a constant of integration then :

A

A =${1 \over {54}}$ and f(x) = 9(x–1)²

B

A =${1 \over {54}}$ and f(x) = 3(x–1)

C

A =${1 \over {81}}$ and f(x) = 3(x–1)

D

A =${1 \over {27}}$ and f(x) = 9(x–1)²

Question 46: Palette 46/76

JEE Mains 2019 MCQ

$\int {{e^{\sec x}}}$ $(\sec x\tan xf(x) + \sec x\tan x + se{x^2}x)dx$
= e^secxf(x) + C then a possible choice of f(x) is :-

A

x sec x + tan x + 1/2

B

sec x + xtan x - 1/2

C

sec x - tan x - 1/2

D

sec x + tan x + 1/2

Question 47: Palette 47/76

JEE Mains 2019 MCQ

The integral $\int {{\rm{se}}{{\rm{c}}^{{\rm{2/ 3}}}}\,{\rm{x }}\,{\rm{cose}}{{\rm{c}}^{{\rm{4 / 3}}}}{\rm{x \,dx}}} $ is equal to (Hence C is a constant of integration)

A

-3/4 tan ^{- 4 / 3} x + C

B

3tan^–1/3x + C

C

–3cot^–1/3x+ C

D

- 3tan^–1/3x + C

Question 48: Palette 48/76

JEE Mains 2019 MCQ

If $\int {{{dx} \over {{x^3}{{(1 + {x^6})}^{2/3}}}} = xf(x){{(1 + {x^6})}^{{1 \over 3}}} + C} $
where C is a constant of integration, then the function ƒ(x) is equal to

A

${3 \over {{x^2}}}$

B

$ - {1 \over {6{x^3}}}$

C

$ - {1 \over {2{x^3}}}$

D

$ - {1 \over {2{x^2}}}$

Question 49: Palette 49/76

JEE Mains 2019 MCQ

$\int {{{\sin {{5x} \over 2}} \over {\sin {x \over 2}}}dx} $ is equal to
(where c is a constant of integration)

A

2x + sinx + 2sin2x + c

B

x + 2sinx + sin2x + c

C

x + 2sinx + 2sin2x + c

D

2x + sinx + sin2x + c

Question 50: Palette 50/76

JEE Mains 2019 MCQ

The integral $\int {{{3{x^{13}} + 2{x^{11}}} \over {{{\left( {2{x^4} + 3{x^2} + 1} \right)}^4}}}} \,dx$ is equal to : (where C is a constant of integration)

A

${{{x^{12}}} \over {6{{\left( {2{x^4} + 3{x^2} + 1} \right)}^3}}}$ + $C$

B

${{{x^4}} \over {6{{\left( {2{x^4} + 3{x^2} + 1} \right)}^3}}} + C$

C

${{{x^{12}}} \over {{{\left( {2{x^4} + 3{x^2} + 1} \right)}^3}}} + C$

D

${{{x^4}} \over {{{\left( {2{x^4} + 3{x^2} + 1} \right)}^3}}} + C$

Question 51: Palette 51/76

JEE Mains 2019 MCQ

The integral $\int \, $cos(log_e x) dx is equal to : (where C is a constant of integration)

A

${x \over 2}$[sin(log_e x) $-$ cos(log_e x)] + C

B

x[cos(log_e x) + sin(log_e x)] + C

C

${x \over 2}$[cos(log_e x) + sin(log_e x)] + C

D

x[cos(log_e x) $-$ sin(log_e x)] + C

Question 52: Palette 52/76

JEE Mains 2019 MCQ

If   $\int {{{x + 1} \over {\sqrt {2x - 1} }}} \,dx$ = f(x) $\sqrt {2x - 1} $ + C, where C is a constant of integration, then f(x) is equal to :

A

${2 \over 3}$ (x $-$ 4)

B

${1 \over 3}$ (x + 4)

C

${1 \over 3}$ (x + 1)

D

${2 \over 3}$ (x + 2)

Question 53: Palette 53/76

JEE Mains 2019 MCQ

If  $\int {{{\sqrt {1 - {x^2}} } \over {{x^4}}}} $ dx = A(x)${\left( {\sqrt {1 - {x^2}} } \right)^m}$ + C, for a suitable chosen integer m and a function A(x), where C is a constant of integration, then (A(x))^m equals :

A

${1 \over {27{x^6}}}$

B

${{ - 1} \over {27{x^9}}}$

C

${1 \over {9{x^4}}}$

D

${1 \over {3{x^3}}}$

Question 54: Palette 54/76

JEE Mains 2019 MCQ

If  $\int \, $x⁵.e^$-$4x3 dx = ${1 \over {48}}$e^$-$4x3 f(x) + C, where C is a constant of inegration, then f(x) is equal to -

A

$-$2x³ $-$ 1

B

$-$ 2x³ + 1

C

4x³ + 1

D

$-$4x³ $-$ 1

Question 55: Palette 55/76

JEE Mains 2019 MCQ

Let n $ \ge $ 2 be a natural number and $0 < \theta < {\pi \over 2}.$ Then $\int {{{{{\left( {{{\sin }^n}\theta - \sin \theta } \right)}^{1/n}}\cos \theta } \over {{{\sin }^{n + 1}}\theta }}} \,d\theta $ is equal to - (where C is a constant of integration)

A

${n \over {{n^2} - 1}}{\left( {1 + {1 \over {{{\sin }^{n - 1}}\theta }}} \right)^{{{n + 1} \over n}}} + C$

B

${n \over {{n^2} - 1}}{\left( {1 - {1 \over {{{\sin }^{n + 1}}\theta }}} \right)^{{{n + 1} \over n}}} + C$

C

${n \over {{n^2} - 1}}{\left( {1 - {1 \over {{{\sin }^{n - 1}}\theta }}} \right)^{{{n + 1} \over n}}} + C$

D

${n \over {{n^2} + 1}}{\left( {1 - {1 \over {{{\sin }^{n - 1}}\theta }}} \right)^{{{n + 1} \over n}}} + C$

Question 56: Palette 56/76

JEE Mains 2019 MCQ

If   $f\left( x \right) = \int {{{5{x^8} + 7{x^6}} \over {{{\left( {{x^2} + 1 + 2{x^7}} \right)}^2}}}} \,dx,\,\left( {x \ge 0} \right),$

$f\left( 0 \right) = 0,$    then the value of $f(1)$ is :

A

$ - $ ${1 \over 2}$

B

$ - $ ${1 \over 4}$

C

${1 \over 2}$

D

${1 \over 4}$

Question 57: Palette 57/76

JEE Mains 2019 MCQ

For x² $ \ne $ n$\pi $ + 1, n $ \in $ N (the set of natural numbers), the integral

$\int {x\sqrt {{{2\sin ({x^2} - 1) - \sin 2({x^2} - 1)} \over {2\sin ({x^2} - 1) + \sin 2({x^2} - 1)}}} dx} $ is equal to :

(where c is a constant of integration)

A

${\log _e}\left| {{1 \over 2}{{\sec }^2}\left( {{x^2} - 1} \right)} \right| + c$

B

${1 \over 2}{\log _e}\left| {\sec \left( {{x^2} - 1} \right)} \right| + c$

C

${1 \over 2}{\log _e}\left| {{{\sec }^2}\left( {{{{x^2} - 1} \over 2}} \right)} \right| + c$

D

${\log _e}\left| {\sec \left( {{{{x^2} - 1} \over 2}} \right)} \right| + c$

Question 58: Palette 58/76

JEE Mains 2018 MCQ

If $\int {{{\tan x} \over {1 + \tan x + {{\tan }^2}x}}dx = x - {K \over {\sqrt A }}{{\tan }^{ - 1}}} $ $\left( {{{K\,\tan x + 1} \over {\sqrt A }}} \right) + C,(C\,\,$ is a constant of integration) then the ordered pair (K, A) is equal to :

A

(2, 1)

B

($-$2, 3)

C

(2, 3)

D

($-$2, 1)

Question 59: Palette 59/76

JEE Mains 2018 MCQ

The integral

$\int {{{{{\sin }^2}x{{\cos }^2}x} \over {{{\left( {{{\sin }^5}x + {{\cos }^3}x{{\sin }^2}x + {{\sin }^3}x{{\cos }^2}x + {{\cos }^5}x} \right)}^2}}}} dx$

is equal to

A

${{ - 1} \over {1 + {{\cot }^3}x}} + C$

B

${1 \over {3\left( {1 + {{\tan }^3}x} \right)}} + C$

C

${{ - 1} \over {3\left( {1 + {{\tan }^3}x} \right)}} + C$

D

${1 \over {1 + {{\cot }^3}x}} + C$

Question 60: Palette 60/76

JEE Mains 2018 MCQ

If    $\int {{{2x + 5} \over {\sqrt {7 - 6x - {x^2}} }}} \,\,dx = A\sqrt {7 - 6x - {x^2}} + B{\sin ^{ - 1}}\left( {{{x + 3} \over 4}} \right) + C$
(where C is a constant of integration), then the ordered pair (A, B) is equal to :

A

(2,  1)

B

($-$ 2,   $-$1)

C

($-$ 2,  1)

D

(2,   $-$1)

Question 61: Palette 61/76

JEE Mains 2018 MCQ

If $f\left( {{{x - 4} \over {x + 2}}} \right) = 2x + 1,$ (x $ \in $ R $-${1, $-$ 2}), then $\int f \left( x \right)dx$ is equal to :
(where C is a constant of integration)

A

12 log_e | 1 $-$ x | + 3x + C

B

$-$ 12 log_e | 1 $-$ x | $-$ 3x + C

C

12 log_e | 1 $-$ x | $-$ 3x + C

D

$-$ 12 log_e | 1 $-$ x | + 3x + C

Question 62: Palette 62/76

JEE Mains 2017 MCQ

If $\,\,\,$ f$\left( {{{3x - 4} \over {3x + 4}}} \right)$ = x + 2, x $ \ne $ $-$ ${4 \over 3}$, and

$\int {} $f(x) dx = A log$\left| {} \right.$1 $-$ x $\left| {} \right.$ + Bx + C,

then the ordered pair (A, B) is equal to :

(where C is a constant of integration)

A

$\left( {{8 \over 3},{2 \over 3}} \right)$

B

$\left( { - {8 \over 3},{2 \over 3}} \right)$

C

$\left( { - {8 \over 3}, - {2 \over 3}} \right)$

D

$\left( { {8 \over 3}, - {2 \over 3}} \right)$

Question 63: Palette 63/76

JEE Mains 2017 MCQ

The integral

$\int {\sqrt {1 + 2\cot x(\cos ecx + \cot x)\,} \,\,dx} $

$\left( {0 < x < {\pi \over 2}} \right)$ is equal to :

(where C is a constant of integration)

A

4 log(sin ${x \over 2}$ ) + C

B

2 log(sin ${x \over 2}$ ) + C

C

2 log(cos ${x \over 2}$ ) + C

D

4 log(cos ${x \over 2}$) + C

Question 64: Palette 64/76

JEE Mains 2017 MCQ

Let ${I_n} = \int {{{\tan }^n}x\,dx} ,\,\left( {n > 1} \right).$

If ${I_4} + {I_6}$ = $a{\tan ^5}x + b{x^5} + C$, where C is a constant of integration,

then the ordered pair $\left( {a,b} \right)$ is equal to

A

$\left( {{1 \over 5},0} \right)$

B

$\left( {{1 \over 5}, - 1} \right)$

C

$\left( { - {1 \over 5},0} \right)$

D

$\left( { - {1 \over 5},1} \right)$

Question 65: Palette 65/76

JEE Mains 2016 MCQ

The integral $\int {{{dx} \over {\left( {1 + \sqrt x } \right)\sqrt {x - {x^2}} }}} $ is equal to :

(where C is a constant of integration.)

A

$ - 2\sqrt {{{1 + \sqrt x } \over {1 - \sqrt x }}} + C$

B

$ - 2\sqrt {{{1 - \sqrt x } \over {1 + \sqrt x }}} + C$

C

$ - \sqrt {{{1 - \sqrt x } \over {1 + \sqrt x }}} + C$

D

$2\sqrt {{{1 + \sqrt x } \over {1 - \sqrt x }}} + C$

Question 66: Palette 66/76

JEE Mains 2016 MCQ

If   $\int {{{dx} \over {{{\cos }^3}x\sqrt {2\sin 2x} }}} = {\left( {\tan x} \right)^A} + C{\left( {\tan x} \right)^B} + k,$

where k is a constant of integration, then A + B +C equals :

A

${{21} \over 5}$

B

${{16} \over 5}$

C

${{7} \over 10}$

D

${{27} \over 10}$

Question 67: Palette 67/76

JEE Mains 2016 MCQ

The integral $\int {{{2{x^{12}} + 5{x^9}} \over {{{\left( {{x^5} + {x^3} + 1} \right)}^3}}}} dx$ is equal to :

A

${{{x^5}} \over {2{{\left( {{x^5} + {x^3} + 1} \right)}^2}}} + C$

B

${{ - {x^{10}}} \over {2{{\left( {{x^5} + {x^3} + 1} \right)}^2}}} + C$

C

${{{-x^5}} \over {{{\left( {{x^5} + {x^3} + 1} \right)}^2}}} + C$

D

${{ {x^{10}}} \over {2{{\left( {{x^5} + {x^3} + 1} \right)}^2}}} + C$

Question 68: Palette 68/76

JEE Mains 2015 MCQ

The integral $\int {{{dx} \over {{x^2}{{\left( {{x^4} + 1} \right)}^{3/4}}}}} $ equals :

A

$ - {\left( {{x^4} + 1} \right)^{{1 \over 4}}} + c$

B

$ - {\left( {{{{x^4} + 1} \over {{x^4}}}} \right)^{{1 \over 4}}} + c$

C

$ {\left( {{{{x^4} + 1} \over {{x^4}}}} \right)^{{1 \over 4}}} + c$

D

$ {\left( {{x^4} + 1} \right)^{{1 \over 4}}} + c$

Question 69: Palette 69/76

JEE Mains 2014 MCQ

The integral $\int {\left( {1 + x - {1 \over x}} \right){e^{x + {1 \over x}}}dx} $ is equal to

A

B

C

D

Question 70: Palette 70/76

JEE Mains 2013 MCQ

If $\int {f\left( x \right)dx = \psi \left( x \right),} $ then $\int {{x^5}f\left( {{x^3}} \right)dx} $ is equal to

A

${1 \over 3}\left[ {{x^3}\psi \left( {{x^3}} \right) - \int {{x^2}\psi \left( {{x^3}} \right)dx} } \right] + C$

B

${1 \over 3}{x^3}\psi \left( {{x^3}} \right) - 3\int {{x^3}\psi \left( {{x^3}} \right)dx} + C$

C

${1 \over 3}{x^3}\psi \left( {{x^3}} \right) - \int {{x^2}\psi \left( {{x^3}} \right)dx} + C$

D

${1 \over 3}\left[ {{x^3}\psi \left( {{x^3}} \right) - \int {{x^3}\psi \left( {{x^3}} \right)dx} } \right] + C$

Question 71: Palette 71/76

JEE Mains 2012 MCQ

If the $\int {{{5\tan x} \over {\tan x - 2}}dx = x + a\,\ln \,\left| {\sin x - 2\cos x} \right| + k,} $ then $a$ is
equal to :

A

$-1$

B

$-2$

C

$1$

D

$2$

Question 72: Palette 72/76

JEE Mains 2008 MCQ

The value of $\sqrt 2 \int {{{\sin xdx} \over {\sin \left( {x - {\pi \over 4}} \right)}}} $ is

A

$\,x + \log \,\left| {\,\cos \left( {x - {\pi \over 4}} \right)\,} \right| + c$

B

$\,x - \log \,\left| {\,\sin \left( {x - {\pi \over 4}} \right)\,} \right| + c$

C

$\,x + \log \,\left| {\,\sin \left( {x - {\pi \over 4}} \right)\,} \right| + c$

D

$\,x - \log \,\left| {\,\cos \left( {x - {\pi \over 4}} \right)\,} \right| + c$

Question 73: Palette 73/76

JEE Mains 2007 MCQ

$\int {{{dx} \over {\cos x + \sqrt 3 \sin x}}} $ equals

A

$\log \,\tan \,\left( {{x \over 2} + {\pi \over {12}}} \right) + C$

B

$\log \,\tan \,\left( {{x \over 2} - {\pi \over {12}}} \right) + C$

C

$\,{1 \over 2}\,\log \,\tan \,\left( {{x \over 2} + {\pi \over {12}}} \right) + C$

D

$\,{1 \over 2}\,\log \,\tan \,\left( {{x \over 2} - {\pi \over {12}}} \right) + C$

Question 74: Palette 74/76

JEE Mains 2005 MCQ

$\int {{{\left\{ {{{\left( {\log x - 1} \right)} \over {1 + {{\left( {\log x} \right)}^2}}}} \right\}}^2}\,\,dx} $ is equal to

A

${{\log x} \over {{{\left( {\log x} \right)}^2} + 1}} + C$

B

${x \over {{x^2} + 1}} + C$

C

${{x{e^x}} \over {1 + {x^2}}} + C$

D

${x \over {{{\left( {\log x} \right)}^2} + 1}} + C$

Question 75: Palette 75/76

JEE Mains 2004 MCQ

$\int {{{dx} \over {\cos x - \sin x}}} $ is equal to

A

${1 \over {\sqrt 2 }}\log \left| {\tan \left( {{x \over 2} + {{3\pi } \over 8}} \right)} \right| + C$

B

${1 \over {\sqrt 2 }}\log \left| {\cot \left( {{x \over 2}} \right)} \right| + C$

C

${1 \over {\sqrt 2 }}\log \left| {\tan \left( {{x \over 2} - {{3\pi } \over 8}} \right)} \right| + C$

D

$\,{1 \over {\sqrt 2 }}\log \left| {\tan \left( {{x \over 2} - {\pi \over 8}} \right)} \right| + C$

Question 76: Palette 76/76

JEE Mains 2004 MCQ

If $\int {{{\sin x} \over {\sin \left( {x - \alpha } \right)}}dx = Ax + B\log \sin \left( {x - \alpha } \right), + C,} $ then value of
$(A, B)$ is

A

$\left( { - \cos \alpha ,\sin \alpha } \right)$

B

$\left( { \cos \alpha ,\sin \alpha } \right)$

C

$\left( { - \sin \alpha ,\cos \alpha } \right)$

D

$\left( { \sin \alpha ,\cos \alpha } \right)$