Indefinite Integrals - Calculus - Mathematics Previous Year Questions

1

2016 · Mathematics · Calculus · Indefinite Integrals

WB JEE 2016

\(\int {{2^x}[f'(x) + f(x)\log 2]dx}\) is equal to

A

2^x f'(x) + C

B

2^x log 2 + C

C

2^x f(x) + C

D

2^x + C

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2

2017 · Mathematics · Calculus · Indefinite Integrals

WB JEE 2017

Let I = \(\left| {\int {_{10}^{19}{{\sin x} \over {1 + {x^8}}}dx} } \right|\). Then,

A

| I | < 10^\(-\)9

B

| I | < 10^\(-\)7

C

| I | < 10^\(-\)5

D

| I | > 10^\(-\)7

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3

2017 · Mathematics · Calculus · Indefinite Integrals

WB JEE 2017

\(\int {\cos (\log x)dx}\) = F(x) + C, where C is an arbitrary constant. Here, F(x) is equal to

A

\(x[\cos (\log x) + \sin (\log x)]\)

B

\(x[\cos (\log x) - \sin (\log x)]\)

C

\({x \over 2}[\cos (\log x) + \sin (\log x)]\)

D

\({x \over 2}[\cos (\log x) - \sin (\log x)]\)

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4

2017 · Mathematics · Calculus · Indefinite Integrals

WB JEE 2017

\(\int {{{{x^2} - 1} \over {{x^4} + 3{x^2} + 1}}dx}\) (x > 0) is

A

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

B

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

C

\({\log _e}\left| {{{x + {1 \over x} - 1} \over {x + {1 \over x} + 1}}} \right| + C\)

D

\({\log _e}\left| {{{x - {1 \over x} - 1} \over {x - {1 \over x} + 1}}} \right| + C\)

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5

2018 · Mathematics · Calculus · Indefinite Integrals

WB JEE 2018

If \(\int {f(x)} \sin x\cos xdx = {1 \over {2({b^2} - {a^2})}}\log (f(x)) + c\), where c is the constant of integration, then f(x) is equal to

A

\({2 \over {({b^2} - {a^2})\sin 2x}}\)

B

\({2 \over {ab\sin 2x}}\)

C

\({2 \over {({b^2} - {a^2})\cos 2x}}\)

D

\({2 \over {ab\cos 2x}}\)

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6

2018 · Mathematics · Calculus · Indefinite Integrals

WB JEE 2018

If \(\int {{e^{\sin x}}} .\left[ {{{x{{\cos }^3}x - \sin x} \over {{{\cos }^2}x}}} \right]dx = {e^{\sin x}}f(x) + c\), where c is constant of integration, then f(x) is equal to

A

sec x \(-\) x

B

x \(-\) sec x

C

tan x \(-\) x

D

x \(-\) tan x

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7

2019 · Mathematics · Calculus · Indefinite Integrals

WB JEE 2019

If \(\int {{2^{{2^x}}}.\,{2^x}dx} = A\,.\,{2^{{2^x}}} + C\), then A is equal to

A

\({1 \over {\log 2}}\)

B

log 2

C

\({{{(\log 2)}^2}}\)

D

\({1 \over {{{(\log 2)}^2}}}\)

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8

2019 · Mathematics · Calculus · Indefinite Integrals

WB JEE 2019

y = \(\int {\cos \left\{ {2{{\tan }^{ - 1}}\sqrt {{{1 - x} \over {1 + x}}} } \right\}} dx\) is an equation of a family of

A

straight lines

B

circles

C

ellipses

D

parabolas

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9

2019 · Mathematics · Calculus · Indefinite Integrals

WB JEE 2019

If \(\int {\cos x\log \left( {\tan {x \over 2}} \right)} dx\) = \(\sin x\log \left( {\tan {x \over 2}} \right)\) + f(x), then f(x) is equal to (assuming c is a arbitrary real constant).

A

c

B

c \(-\) x

C

c + x

D

2x + c

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10

2020 · Mathematics · Calculus · Indefinite Integrals

WB JEE 2020

$$\int {{{f(x)\phi '(x) + \phi (x)f'(x)} \over {(f(x)\phi (x) + 1)\sqrt {f(x)\phi (x) - 1} }}dx = }$$

A

\({\sin ^{ - 1}} = \sqrt {{{f(x)} \over {\phi (x)}}} + c\)

B

\({\cos ^{ - 1}}\sqrt {{{(f(x))}^2} - {{(\phi (x))}^2}} + c\)

C

\(\sqrt 2 {\tan ^{ - 1}}\sqrt {{{f(x)\phi (x) - 1} \over 2}} + c\)

D

\(\sqrt 2 {\tan ^{ - 1}}\sqrt {{{f(x)\phi (x) + 1} \over 2}} + c\)

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11

2021 · Mathematics · Calculus · Indefinite Integrals

WB JEE 2021

If \(\int {{{\sin 2x} \over {{{(a + b\cos x)}^2}}}dx} = \alpha \left[ {{{\log }_e}\left| {a + b\cos x} \right| + {a \over {a + b\cos x}}} \right] + c\), then \(\alpha\) is equal to

A

\({2 \over {{b^2}}}\)

B

\({2 \over {{a^2}}}\)

C

\(- {2 \over {{b^2}}}\)

D

\(- {2 \over {{a^2}}}\)

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12

2022 · Mathematics · Calculus · Indefinite Integrals

WB JEE 2022

Let \(\int {{{{x^{{1 \over 2}}}} \over {\sqrt {1 - {x^3}} }}dx = {2 \over 3}g(f(x)) + c}\) ; then

(c denotes constant of integration)

A

\(f(x) = \sqrt x ,g(x) = {x^{{3 \over 2}}}\)

B

\(f(x) = {x^{{3 \over 2}}},g(x) = {\sin ^{ - 1}}x\)

C

\(f(x) = \sqrt x ,g(x) = {\sin ^{ - 1}}x\)

D

\(f(x) = {\sin ^{ - 1}}x,g(x) = {x^{{3 \over 2}}}\)

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13

2022 · Mathematics · Calculus · Indefinite Integrals

WB JEE 2022

\(I = \int {\cos (\ln x)dx}\). Then I =

A

\({x \over 2}\{ \cos (\ln x) + \sin (\ln x)\} + c\) (c denotes constant of integration)

B

\({x^2}\{ \cos (\ln x) - \sin (\ln x)\} + c\) (c denotes constant of integration)

C

\({x^2}\sin (\ln x) + c\) (c denotes constant of integration)

D

\(x\cos (\ln x) + c\) (c denotes constant of integration)

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14

2023 · Mathematics · Calculus · Indefinite Integrals

WB JEE 2023

If \(I = \int {{{{x^2}dx} \over {{{(x\sin x + \cos x)}^2}}} = f(x) + \tan x + c}\), then \(f(x)\) is

A

\({{\sin x} \over {x\sin x + \cos x}}\)

B

\({1 \over {{{(x\sin x + \cos x)}^2}}}\)

C

\({{ - x} \over {\cos x(x\sin x + \cos x)}}\)

D

\({1 \over {\sin x(x\cos x + \sin x)}}\)

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15

2023 · Mathematics · Calculus · Indefinite Integrals

WB JEE 2023

If \(\int {{{dx} \over {(x + 1)(x - 2)(x - 3)}} = {1 \over k}{{\log }_e}\left\{ {{{|x - 3{|^3}|x + 1|} \over {{{(x - 2)}^4}}}} \right\} + c}\), then the value of k is

A

4

B

6

C

8

D

12

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16

2024 · Mathematics · Calculus · Indefinite Integrals

WB JEE 2024

$$\text { If } \int \frac{\log _e\left(x+\sqrt{1+x^2}\right)}{\sqrt{1+x^2}} \mathrm{~d} x=\mathrm{f}(\mathrm{g}(x))+\mathrm{c} \text { then }$$

A

\(\mathrm{f}(x)=\frac{x^2}{2}, \mathrm{~g}(x)=\log _{\mathrm{e}}\left(x+\sqrt{1+x^2}\right)\)

B

\(\mathrm{f}(x)=\log _{\mathrm{e}}\left(x+\sqrt{1+x^2}\right), \mathrm{g}(x)=\frac{x^2}{2}\)

C

\(\mathrm{f}(x)=x^2, \mathrm{~g}(x)=\log _{\mathrm{e}}\left(x+\sqrt{1+x^2}\right)\)

D

\(\mathrm{f}(x)=\log _{\mathrm{e}}\left(x-\sqrt{1+x^2}\right), \mathrm{g}(x)=x^2\)

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17

2008 · Mathematics · Calculus · Indefinite Integrals

WB JEE 2008

Evaluate \(\int {{{{x^2}} \over {x(1 + {x^2})}}dx}\)

Write your response

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18

2009 · Mathematics · Calculus · Indefinite Integrals

WB JEE 2009

\(\int {{{dx} \over {x(x + 1)}}}\) equals

where c is arbitrary constant.

A

\(\ln \left| {{{x + 1} \over x}} \right| + c\)

B

\(\ln \left| {{x \over {x + 1}}} \right| + c\)

C

\(\ln \left| {{{x - 1} \over x}} \right| + c\)

D

\(\ln \left| {{{x - 1} \over {x + 1}}} \right| + c\)

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19

2009 · Mathematics · Calculus · Indefinite Integrals

WB JEE 2009

The value of \(\int\limits_{ - 1}^1 {{{|x + 2|} \over {x + 2}}dx}\) is

A

1

B

2

C

0

D

\(-\)1

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20

2009 · Mathematics · Calculus · Indefinite Integrals

WB JEE 2009

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

where c is an arbitrary constant

A

\(\log ({\sin ^{ - 1}}x) + c\)

B

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

C

\(\log \left( {\sqrt {1 - {x^2}} } \right) + c\)

D

\(\sin ({\cos ^{ - 1}}x) + c\)

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