1
MCQ (Single Correct Answer)

JEE Main 2019 (Online) 10th January Morning Slot

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

Explanation

$$\int {{{{{\left( {{{\sin }^n}\theta - \sin \theta } \right)}^{1/n}}\cos \theta } \over {{{\sin }^{n + 1}}\theta }}} \,d\theta $$

$$ = \int {{{\sin \theta {{\left( {1 - {1 \over {{{\sin }^{n - 1}}\theta }}} \right)}^{1/n}}} \over {{{\sin }^{n + 1}}\theta }}} \,d\theta $$

Put $$1 - {1 \over {{{\sin }^{n - 1}}\theta }} = t$$

So $${{\left( {n - 1} \right)} \over {{{\sin }^n}\theta }}\cos \theta d\theta = dt$$

Now  $${1 \over {n - 1}}\int {{{\left( t \right)}^{{1 \over n} + 1}}dt} $$

$$ = {1 \over {\left( {n - 1} \right)}}{{{{\left( t \right)}^{{1 \over n} + 1}}} \over {{1 \over n} + 1}} + C$$

$$ = {n \over {\left( {n^2 - 1} \right)}}{\left( {1 - {1 \over {{{\sin }^{n - 1}}\theta }}} \right)^{{1 \over n} + 1}} + C$$
2
MCQ (Single Correct Answer)

JEE Main 2019 (Online) 10th January Evening Slot

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

Explanation

$$\int {{x^5}} .{e^{ - 4{x^3}}}\,dx = {1 \over {48}}{e^{ - 4{x^3}}}f\left( x \right) + c$$

Put  $${x^3} = t$$

$$3{x^2}\,dx = dt$$

$$\int {{x^3}.{e^{ - 4{x^3}}}.\,{x^2}} dx$$

$${1 \over 3}\int {t.{e^{ - 4t}}dt} $$

$${1 \over 3}\left[ {t.{{{e^{ - 4t}}} \over { - 4}} - \int {{{{e^{ - 4t}}} \over { - 4}}dt} } \right]$$

$$ - {{{e^{ - 4t}}} \over {48}}\left[ {4t + 1} \right] + c$$

$${{ - {e^{ - 4{x^3}}}} \over {48}}\left[ {4{x^3} + 1} \right] + c$$

$$ \therefore $$  $$f(x) = - 1 - 4{x^3}$$
3
MCQ (Single Correct Answer)

JEE Main 2019 (Online) 11th January Evening Slot

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)

Explanation

$$\sqrt {2x - 1} = t \Rightarrow 2x - 1 = {t^2} \Rightarrow 2dx = 2t.dt$$

$$\int {{{x + 1} \over {\sqrt {2x - 1} }}dx = \int {{{{{{t^2} + 1} \over 2} + 1} \over t}tdt = \int {{{{t^2} + 3} \over 2}dt} } } $$

$$ = {1 \over 2}\left( {{{{t^3}} \over 3} + 3t} \right) = {t \over 6}\left( {{t^2} + 9} \right) + c$$

$$ = \sqrt {2x - 1} \left( {{{2x - 1 + 9} \over 6}} \right) + c = \sqrt {2x - 1} \left( {{{x + 4} \over 3}} \right) + c$$

$$ \Rightarrow f\left( x \right) = {{x + 4} \over 3}$$
4
MCQ (Single Correct Answer)

JEE Main 2019 (Online) 12th January Morning Slot

The integral $$\int \, $$cos(loge x) dx is equal to : (where C is a constant of integration)
A
$${x \over 2}$$[sin(loge x) $$-$$ cos(loge x)] + C
B
x[cos(loge x) + sin(loge x)] + C
C
$${x \over 2}$$[cos(loge x) + sin(loge x)] + C
D
x[cos(loge x) $$-$$ sin(loge x)] + C

Explanation

$${\rm I} = \int {\cos \left( {\ell nx} \right)} dx$$

$${\rm I} = \cos (\ln x).x + \int {\sin \left( {\ell nx} \right)dx} $$

$$\cos \left( {\ell nx} \right)x + \left[ {\sin \left( {\ell nx} \right).x - \int {\cos \left( {\ell nx} \right)dx} } \right]$$

$${\rm I} = {x \over 2}\left[ {\sin \left( {\ell nx} \right) + \cos \left( {\ell nx} \right)} \right] + C$$

Questions Asked from Indefinite Integrals

On those following papers in MCQ (Single Correct Answer)
Number in Brackets after Paper Name Indicates No of Questions
AIEEE 2004 (2)
keyboard_arrow_right
AIEEE 2005 (1)
keyboard_arrow_right
AIEEE 2007 (1)
keyboard_arrow_right
AIEEE 2008 (1)
keyboard_arrow_right
AIEEE 2012 (1)
keyboard_arrow_right
JEE Main 2013 (Offline) (1)
keyboard_arrow_right
JEE Main 2014 (Offline) (1)
keyboard_arrow_right
JEE Main 2015 (Offline) (1)
keyboard_arrow_right
JEE Main 2016 (Offline) (1)
keyboard_arrow_right
JEE Main 2016 (Online) 9th April Morning Slot (1)
keyboard_arrow_right
JEE Main 2016 (Online) 10th April Morning Slot (1)
keyboard_arrow_right
JEE Main 2017 (Offline) (1)
keyboard_arrow_right
JEE Main 2017 (Online) 8th April Morning Slot (1)
keyboard_arrow_right
JEE Main 2017 (Online) 9th April Morning Slot (1)
keyboard_arrow_right
JEE Main 2018 (Offline) (1)
keyboard_arrow_right
JEE Main 2018 (Online) 15th April Morning Slot (1)
keyboard_arrow_right
JEE Main 2018 (Online) 15th April Evening Slot (1)
keyboard_arrow_right
JEE Main 2018 (Online) 16th April Morning Slot (1)
keyboard_arrow_right
JEE Main 2019 (Online) 9th January Morning Slot (1)
keyboard_arrow_right
JEE Main 2019 (Online) 9th January Evening Slot (1)
keyboard_arrow_right
JEE Main 2019 (Online) 10th January Morning Slot (1)
keyboard_arrow_right
JEE Main 2019 (Online) 10th January Evening Slot (1)
keyboard_arrow_right
JEE Main 2019 (Online) 11th January Evening Slot (1)
keyboard_arrow_right
JEE Main 2019 (Online) 12th January Morning Slot (1)
keyboard_arrow_right
JEE Main 2019 (Online) 12th January Evening Slot (1)
keyboard_arrow_right
JEE Main 2019 (Online) 8th April Morning Slot (1)
keyboard_arrow_right
JEE Main 2019 (Online) 8th April Evening Slot (1)
keyboard_arrow_right
JEE Main 2019 (Online) 9th April Morning Slot (1)
keyboard_arrow_right
JEE Main 2019 (Online) 9th April Evening Slot (1)
keyboard_arrow_right
JEE Main 2019 (Online) 10th April Morning Slot (1)
keyboard_arrow_right
JEE Main 2019 (Online) 10th April Evening Slot (1)
keyboard_arrow_right
JEE Main 2019 (Online) 12th April Morning Slot (1)
keyboard_arrow_right
JEE Main 2019 (Online) 12th April Evening Slot (1)
keyboard_arrow_right
JEE Main 2020 (Online) 8th January Morning Slot (1)
keyboard_arrow_right
JEE Main 2020 (Online) 9th January Morning Slot (2)
keyboard_arrow_right
JEE Main 2020 (Online) 9th January Evening Slot (1)
keyboard_arrow_right
JEE Main 2020 (Online) 3rd September Evening Slot (1)
keyboard_arrow_right
JEE Main 2020 (Online) 4th September Morning Slot (1)
keyboard_arrow_right
JEE Main 2020 (Online) 5th September Morning Slot (1)
keyboard_arrow_right
JEE Main 2020 (Online) 5th September Evening Slot (1)
keyboard_arrow_right
JEE Main 2021 (Online) 24th February Morning Slot (1)
keyboard_arrow_right
JEE Main 2021 (Online) 25th February Morning Slot (1)
keyboard_arrow_right
JEE Main 2021 (Online) 25th February Evening Shift (1)
keyboard_arrow_right
JEE Main 2021 (Online) 18th March Morning Shift (1)
keyboard_arrow_right
JEE Main 2021 (Online) 31st August Morning Shift (1)
keyboard_arrow_right

EXAM MAP

Joint Entrance Examination

JEE Advanced JEE Main

Graduate Aptitude Test in Engineering

GATE CSE GATE ECE GATE ME GATE PI GATE EE GATE CE GATE IN

Medical

NEET