1
MCQ (Single Correct Answer)

JEE Main 2016 (Online) 9th April Morning Slot

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
$${{27} \over 10}$$

Explanation

$$\int {{{dx} \over {{{\cos }^3}x\sqrt {2\sin 2x} }}} $$

=  $$\int {{{dx} \over {{{\cos }^3}x\sqrt {4\sin x\cos x} }}} $$

=  $$\int {{{dx} \over {2{{\cos }^4}x\sqrt {\tan x} }}} $$

Let tan x   =   t2

$$ \Rightarrow $$$$\,\,\,$$ sec2xdx = 2t dt

as   sec2x = 1 + tan2x = 1 + t4

=  $$\int {{{{{\sec }^4}x\,dx} \over {2\sqrt {\tan x} }}} $$

=  $$\int {{{{{\sec }^2}x\left( {{{\sec }^2}x\,dx} \right)} \over {2\sqrt {\tan x} }}} $$

=  $$\int {{{\left( {1 + {t^4}} \right)2t\,dt} \over {2t}}} $$

=   $$\int {\left( {1 + {t^4}} \right)} \,dt$$

=  t + $${{{t^5}} \over 5}$$ + k

=   $$\sqrt {\tan x} $$ + $${1 \over 5}$$ tan$$^{{5 \over 2}}$$x + k

By comparing with the given equation, we get

A = $${1 \over 2}$$, B = $${5 \over 2}$$, C = $${1 \over 5}$$

$$\therefore\,\,\,$$ A + B + C = $${{16} \over 5}$$
2
MCQ (Single Correct Answer)

JEE Main 2016 (Online) 10th April Morning Slot

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

Explanation

I   =   $$\int {{{dx} \over {\left( {1 + \sqrt x } \right)\sqrt {x - {x^2}} }}} $$

=  $$\int {{{dx} \over {\left( {1 + \sqrt x } \right)\sqrt x \sqrt {1 - x} }}} $$

Let  1 + $$\sqrt x $$ = t

$$ \Rightarrow $$$$\,\,\,$$$${1 \over {2\sqrt x }}\,dx$$ = dt

I  =  $$\int {{{2dt} \over {t\sqrt {2t - {t^2}} }}} $$

Again let t = $${1 \over z}$$

$$ \Rightarrow $$$$\,\,\,$$ dt = $$-$$ $${1 \over {{z^2}}}dz$$

$$\therefore\,\,\,$$ I = 2 $$\int {{{ - {1 \over {{z^2}}}dz} \over {{1 \over z}\sqrt {{2 \over z} - {1 \over {{z^2}}}} }}} $$

=  2$$\int {{{ - dz} \over {\sqrt {2z - 1} }}} $$

=  $$ - 2\sqrt {2z - 1} + C$$

=   $$ - 2\sqrt {{2 \over t} - 1} + C$$

=  $$- 2\sqrt {{{2 - t} \over t}} + C$$

=  $$ - 2\sqrt {{{1 - \sqrt x } \over {1 + \sqrt x }}} + C$$
3
MCQ (Single Correct Answer)

JEE Main 2017 (Offline)

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

Explanation

Given,

In = $$\int {{{\tan }^n}x\,dx,\,\,\,n > 1} $$

$$\therefore\,\,\,$$ I4 = $$\int {{{\tan }^4}x\,dx} $$

and I6 = $$\int {{{\tan }^6}} x\,dx$$

$$\therefore\,\,\,$$ I = I4 + I6

= $$\int {\left( {{{\tan }^4}x + {{\tan }^6}x} \right)} dx$$

= $$\int {{{\tan }^4}} x\left( {1 + {{\tan }^2}x} \right)dx$$

= $$\int {{{\tan }^4}} x.{\sec ^2}x\,dx$$

Let, tanx = t

$$ \Rightarrow $$$$\,\,\,$$ sec2x dx = dt

$$\therefore\,\,\,$$ I = $$\int {{t^4}\,dt} $$

= $${1 \over 5}$$ t5 + C

= $${1 \over 5}$$ tan5x + C

$$\therefore\,\,\,$$ By comparing with the question, we get

A = $${1 \over 5}$$,  B = 0
4
MCQ (Single Correct Answer)

JEE Main 2017 (Online) 8th April Morning Slot

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

Explanation

Let, I = $$\int {\sqrt {1 + 2\cot x\cos ec + 2{{\cot }^2}x} .dx} $$

$$ \Rightarrow $$ I = $$\int {\sqrt {{{{{\sin }^2}x + 2\cos x + 2{{\cos }^2}x} \over {{{\sin }^2}x}}} .dx} $$

$$ \Rightarrow $$ I = $$\int {\sqrt {{{1 + 2\cos x + {{\cos }^2}x} \over {\sin x}}} .dx} $$

$$ \Rightarrow $$ I = $$\int {\left| {{{1 + \cos x} \over {\sin x}}} \right|dx} $$

$$ \Rightarrow $$ I = $$\int {\left| {\cos ec\,x + \cot x} \right|.dx} $$

$$ \Rightarrow $$ I = $$\log \left| {\cos ec\,x - \cot x} \right| + \log \left| {\sin x} \right| + C$$

$$ \Rightarrow $$ I = $$\log \left| {1 - \cos x} \right| + C$$

$$ \Rightarrow $$ I = $$\log \left| {2{{\sin }^2}{x \over 2}} \right| + C$$

$$ \Rightarrow $$ I = $$\log \left| {{{\sin }^2}{x \over 2}} \right| + \log 2+ C$$

$$ \Rightarrow $$ I = 2$$\log \left| {{{\sin }}{x \over 2}} \right| + C_1$$

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