If $\mathop \smallint \limits_{\pi /2}^x \,\,\sqrt {3\, - \,2\,\,{{\sin }^2}\,\,u} \,du\,\, + \,\,\mathop \smallint \nolimits_0^y \,\,\cos \,\,t\,dt = 0$ then, $\frac{{dy}}{{dx}}$ is equal to

$\frac{{\sqrt {4 - 3\,\,{{\sin }^2}\,\,x} }}{{\cos \,\,y}}$

Let $f\,\left( x \right)$ be a function satisfying $f'\left( x \right) = f\left( x \right)$ with $f\left( 0 \right) = 1$ and ${\rm{g}}\,\,\left( x \right)$ be a function that satisfies $f\left( x \right)\,\, + \,\,{\rm{g}}\left( x \right) = {x^2}$. Then, the value of the integral $\mathop \smallint \nolimits_0^1 \,\,f\left( x \right)\,\,{\rm{g}}\,\,\left( x \right)\,\,dx$, is

$e - \frac{{{e^2}}}{2} - \frac{5}{2}$

$e\,\, + \,\,\frac{{{e^2}}}{2}\,\, + \,\,\frac{5}{2}$

$\mathop \smallint \limits_1^{\frac{{4\sqrt 3 }}{5} - 1} \,\,\frac{{x + 2}}{{\sqrt {{x^2} + 2x - 3} }}\,\,dx$ equals to

$\frac{{2\,\sqrt 3 }}{3}\,\, - \,\,\frac{1}{2}\,\,\log \,\,\left( {\sqrt 3 + 2} \right)$

$\frac{{2\,\sqrt 3 }}{3}\,\, + \,\,\frac{1}{2}\,\,\log \,\,\left( {\sqrt 3 + 2} \right)$

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JEE • MATHEMATICS • DEFINITE INTEGRATION

Definite integration practice · Hard

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These sets use single-correct-answer MCQs. They do not cover every JEE Main or JEE Advanced question format and are not full-length mock papers.

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Showing 1-30 of 51 questions

1 Hard

1 of 51

Application +4 | -1

For each positive integer

n

, define a function

f_n

[0,1]

as follows:

f_n(x)=\left\{\begin{array}{ccc} 0 & \text {if} & x=0 \\ \sin \frac{\pi}{2 n} & \text {if} & 0 < x \leq \frac{1}{n} \\ \sin \frac{2 \pi}{2 n} & \text {if} & \frac{1}{n} < x \leq \frac{2}{n} \\ \sin \frac{3 \pi}{2 n} & \text {if} & \frac{2}{n} < x \leq \frac{3}{n} \\ \sin \frac{n \pi}{2 n} & \text {if} & \frac{n-1}{n} < x \leq 1 \end{array}\right.

Then the value of

\lim \limits_{{n \rightarrow \infty}} \int\limits_{0}^1 f_n(x) d x

is:

View answer and explanation

Correct option: D

\frac{2}{\pi}

Option D (

\frac{2}{\pi}

) is correct because the limit of the integral is evaluated by recognizing it as a Riemann sum and converting it to a definite integral.

2 Hard

2 of 51

Application +4 | -1

The value of

\int\limits_ {e^{-1}}^{e^2}\left|\frac{\log _e x}{x}\right| d x

View answer and explanation

Correct option: B

\frac{5}{2}

Option B (

\frac{5}{2}

) is correct because by splitting the integral based on the sign of

\log_e x

and using substitution, the integral evaluates to

\frac{5}{2}

3 Hard

3 of 51

Application +4 | -1

The value of

\int\limits_{{-1}}^1 \frac{d}{d x}\left(\tan ^{-1} \frac{1}{x}\right) d x

, is

View answer and explanation

Correct option: C

-\pi / 2

Option C (-

\frac{\pi}{2}

) is correct because the derivative of

\tan^{-1}(\frac{1}{x})

is -

\frac{1}{1+x^2}

, and integrating this from -1 to 1 gives -

\frac{\pi}{2}

4 Hard

4 of 51

Application +4 | -1

\lim \limits_{{n \rightarrow \infty}} \frac{1}{\sqrt{n} \sqrt{n+1}}+\frac{1}{\sqrt{n} \sqrt{n+2}}+\ldots +\frac{1}{\sqrt{n} \sqrt{4 n}}

is equal to:

View answer and explanation

Correct option: A

Option A (2) is correct because the limit of the Riemann sum can be converted to a definite integral that evaluates to 2.

5 Hard

5 of 51

Application +4 | -1

The value of

\int\limits_{{-20 \pi}}^{20 \pi}[\sin x+\cos x] d x

is: (where [.] denotes greatest integer function)

View answer and explanation

Correct option: B

-20 \pi

Option B (

-20 \pi

) is correct because the integral of the greatest integer function of

\sin x + \cos x

over the given interval evaluates to

-20\pi

6 Hard

6 of 51

Application +4 | -1

\int \limits_{\pi /2}^x \,\,\sqrt {3\, - \,2\,\,{{\sin }^2}\,\,u} \,du\,\, + \,\,\int \nolimits_0^y \,\,\cos \,\,t\,dt = 0

then,

\frac{{dy}}{{dx}}

is equal to

View answer and explanation

Correct option: B

- \,\,\frac{{\sqrt {3\,\, - \,\,2\,\,{{\sin }^2}\,\,x} }}{{\cos \,\,y}}

Option B is correct because differentiating the given equation with respect to x and solving for dy/dx yields the stated result.

7 Hard

7 of 51

Application +4 | -1

Let

f(x)=\int\limits_{{-2}}^x e^{(1+t)^2} d t

and

g(x)=f(h(x))

where

h(x)

is defined for all

x \in R

. If

g^{\prime}(2)=e^4

and

h^{\prime}(2)=

1 then absolute value of sum of all possible values of

h(2)

View answer and explanation

Correct option: A

Option A (2) is correct because it represents the absolute value of the sum of all possible values of h(2) derived from the given conditions and differentiation.

8 Hard

8 of 51

Analyzing +4 | -1

The value of the integral

\int_0^{\pi / 2} \sin 2 n x \cot x d x

, where

n

is a positive integer, is

View answer and explanation

Correct option: A

\frac{\pi}{2}

Option A (

\frac{\pi}{2}

) is correct because the integral evaluates to

\frac{\pi}{2}

after using trigonometric identities and properties of definite integrals.

9 Hard

9 of 51

Application +4 | -1

Let

f\,\left( x \right)

be a function satisfying

f'\left( x \right) = f\left( x \right)

with

f\left( 0 \right) = 1

and

\mathrm{g}\,\,\left( x \right)

be a function that satisfies

f\left( x \right)\,\, + \,\,\mathrm{g}\left( x \right) = {x^2}

. Then, the value of the integral

\int \nolimits_0^1 \,\,f\left( x \right)\,\,\mathrm{g}\,\,\left( x \right)\,\,dx

, is

View answer and explanation

Correct option: C

e\,\, - \,\,\frac{{{e^2}}}{2}\,\, - \,\,\frac{3}{2}

Option C (

e\, - \,\frac{{{e^2}}}{2}\, - \,\frac{3}{2}

) is correct because it is the result of integrating the product of the two functions after finding f(x) and g(x).

10 Hard

10 of 51

Analyzing +4 | -1

Let

f\;:R \to R

be a continuous function such that

f\left( x \right)

is not identically equal to zero. If

\int \limits_2^5 \,\,f\left( x \right)\,\,dx = \int \limits_{ - 2}^5 \,\,f\left( x \right)\,\,dx

, then

f\left( x \right)

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the given condition only implies that the integral from -2 to 2 is zero, which doesn't necessarily mean the function is odd or even.

11 Hard

11 of 51

Analyzing +4 | -1

\alpha=\int\limits_{0}^1 e^{9 x+3 \tan ^{-1} x}\left(\frac{12+9 x^2}{1+x^2}\right) d x

, where

\tan ^{-1} x

takes only principal values, then the value of

\left(\log _e|1+\alpha|-\frac{3 \pi}{4}\right)

, is

View answer and explanation

Correct option: A

Option A (9) is correct because it is found by substituting and simplifying the given integral and then taking the logarithm.

12 Hard

12 of 51

Analyzing +4 | -1

Let

f:[-1,2] \rightarrow[0, \infty)

be a continuous function such that

f(x)=f(1-x)

for all

x \in[-1,2]

Let

R_1=\int\limits_{{-1}}^2 x f(x) \,d x

; and

R_2

be the area of the region bounded by

y=f(x), x=-1, x=2

and the

x

-axis. Then,

View answer and explanation

Correct option: C

2 R_1=R_2

Option C (

2 R_1=R_2

) is correct because by using the property of definite integrals and the given condition

f(x) = f(1-x)

, we can relate

R_1

and

R_2

13 Hard

13 of 51

Application +4 | -1

f:[-1,1] \rightarrow\left[-\frac{1}{2}, \frac{1}{2}\right], f(x)=\frac{x}{1+x^2}

. The area bounded by

y=f^{-1}(x), X

-axis,

x=\frac{1}{2}, x=-\frac{1}{2}

View answer and explanation

Correct option: B

\log \left(\frac{e}{2}\right)

Option B (

\log(\frac{e}{2})

) is correct because it represents the area bounded by the inverse function, X-axis, and the given limits.

14 Hard

14 of 51

Analyzing +4 | -1

\int \limits_{ - \pi /2}^{\pi /2} \,\,{\log _e}\,\,\left\{ {\left( {\frac{{a{x^2} + \,\,bx + c}}{{a{x^2} - \,\,bx + c}}} \right)\,\,\left( {a + b} \right)\,\,\left| {\,\sin \,\,x\,} \right|} \right\}\,\,dx

is equal to

View answer and explanation

Correct option: B

\pi \,\,{\log _e}\,\,\left( {\frac{{a + b}}{2}} \right)

Option B (

\pi \,{\log _e}\,\left( {\frac{{a + b}}{2}} \right)

) is correct because the integral simplifies using properties of logarithms and even/odd functions.

15 Hard

15 of 51

Analyzing +4 | -1

f\left( x \right) = \frac{{{e^x}}}{{1\,\, + \,\,{e^x}}},\,\,{I_1} = \int \nolimits_{f\left( { - a} \right)}^{f\left( a \right)} \,\,x\mathrm{g}\,\,\left[ {x\,\,\left( {1 - x} \right)} \right]dx\,\,\;\mathrm{and}

and

{I_2} = \int \nolimits_{f\left( { - a} \right)}^{f\left( a \right)} \,\,\mathrm{g}\,\,\left[ {x\,\left( {1 - x} \right)} \right]\,\,dx

, then the value of

\frac{{{I_2}}}{{{I_1}}}

View answer and explanation

Correct option: A

\mathrm{2}

Option A (2) is correct because by using the property of definite integrals and the given function, the ratio simplifies to 2.

Explore Other Chapters in MATHEMATICS

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SEQUENCE OF SERIES 1764 MCQs MATRICES 1363 MCQs INDEFINITE INTEGRATION 1179 MCQs PROBABILITY 1146 MCQs COMPLEX NUMBERS 980 MCQs DIFFERENTIAL EQUATIONS 977 MCQs

16 Hard

16 of 51

Application +4 | -1

If for all real triplets

(a,\,b,\,c)

f\left( x \right) = a + bx + c{x^2};

then

\int_0^1 {f\left( x \right)dx}

is equal to

View answer and explanation

Correct option: D

\frac{1}{6}\left\{ {f\left( 0 \right) + f\left( 1 \right) + 4f\left( {\frac{1}{2}} \right)} \right\}

Option D is correct because the integral of the given quadratic function can be expressed in terms of the function's values at specific points, matching Simpson's 1/3 rule.

17 Hard

17 of 51

Analyzing +4 | -1

Let

f:[0,5] \rightarrow R

such that

f(1)=3

and

g(x)=\int_1^x t f(t) d t

.If

F(x)=\int_0^x x^2 g(x)

, then

F(x)

has

View answer and explanation

Correct option: A

a local minimum at

x=1

Option A is correct because F''(1) > 0, indicating a local minimum at x=1 after applying the fundamental theorem of calculus and given conditions.

18 Hard

18 of 51

Application +4 | -1

\int\limits_ {\log \sqrt{\pi / 2}}^{\log \sqrt{\pi}} e^{2 x} \sec ^2\left(\frac{1}{3} e^{2 x}\right) d x

is equal to

View answer and explanation

Correct option: A

\sqrt{3}

Option A is correct because the definite integral evaluates to sqrt(3) after applying a u-substitution and evaluating the resulting trigonometric integral.

19 Hard

19 of 51

Application +4 | -1

\int\limits_\pi^{\frac{3 \pi}{2}}[2 \sin x] d x=\frac{-5 \pi}{k}

(where [.] is GIF) then

\mathrm{k}=

View answer and explanation

Correct option: A

Option A (6) is correct because the integral evaluates to

\frac{{-5 \pi}}{{6}}

, thus k = 6.

20 Hard

20 of 51

Application +4 | -1

The value of the integral

\int_0^1 x \cot ^{-1}\left(1-x^2+x^4\right) d x

View answer and explanation

Correct option: A

\frac{\pi}{4}-\frac{1}{2} \log _{\mathrm{e}} 2

Option A (

\frac{{\pi}}{{4}}-\frac{{1}}{{2}} \log _{\mathrm{{e}}} 2

) is correct because the integral is solved by using trigonometric identities and substitution.

21 Hard

21 of 51

Application +4 | -1

For any real number

x

, the value of

\int\limits_{0}^x[x] d x

, is

View answer and explanation

Correct option: C

x[x]-\frac{1}{2}[x]([x]+1)

Option C (

x[x]-\frac{1}{2}[x]([x]+1)

) is correct because it represents the general form of the integral of the greatest integer function from 0 to x.

22 Hard

22 of 51

Application +4 | -1

\int \limits_1^{\frac{{4\sqrt 3 }}{5} - 1} \,\,\frac{{x + 2}}{{\sqrt {{x^2} + 2x - 3} }}\,\,dx

equals to

View answer and explanation

Correct option: B

\frac{{2\,\sqrt 3 }}{3}\,\, + \,\,\frac{1}{2}\,\,\log \,\,3

Option B (

\frac{{2\,\sqrt 3 }}{3}\,\,+\,\,\frac{1}{2}\,\,\log \,3

) is correct because it is the result of evaluating the definite integral after proper substitution and simplification.

23 Hard

23 of 51

Analyzing +4 | -1

The areas of the figure into which curve

{y^2} = 6x

divides the circle

{x^2} + {y^2} = 16

are in the ratio

View answer and explanation

Correct option: C

\frac{{4\;\pi + \sqrt 3 }}{{8\;\pi - \sqrt 3 }}

Option C (

\frac{{4\;\pi + \sqrt 3 }}{{8\;\pi - \sqrt 3 }}

) is correct because it represents the ratio of the area bounded by the parabola and the circle to the remaining area of the circle.

24 Hard

24 of 51

Analyzing +4 | -1

\frac{\int_{-x}^x f(t) d t}{\int_0^{2 x} f(t+4) d t}

is equal to

View answer and explanation

Correct option: D

\frac{f(0)}{f(4)}

Option D (f(0)/f(4)) is correct because L'Hopital's rule is applied to evaluate the limit.

25 Hard

25 of 51

Analyzing +4 | -1

The value of

\int \limits_{ - 1}^1 \,\,{\sin ^{ - 1}}\,\,\left[ {{x^2}\,\, + \,\,\frac{1}{2}} \right]\,\,dx\,\, + \,\,\int \limits_{ - 1}^1 \,\,{\cos ^{ - 1}}\,\,\left[ {{x^2} - \,\,\frac{1}{2}} \right]\,\,dx

, where

\left[ {\, \cdot \,} \right]

denotes the greatest integer function, is

View answer and explanation

Correct option: B

2\,\pi

Option B (2π) is correct because the sum of the two integrals simplifies to 2π after evaluating the greatest integer functions and using trigonometric identities.

26 Hard

26 of 51

Application +4 | -1

\int\limits_ {\frac{\pi}{2}}^x \sqrt{3-2 \sin ^2 t} d t+\int\limits_{0}^y \cos t d t=0

then

\left(\frac{d y}{d x}\right)_{(\pi, \pi)}

is equal to -

View answer and explanation

Correct option: B

\sqrt{3}

Option B (

\sqrt{3}

) is correct because it is the value of the derivative dy/dx at the specified point, obtained through implicit differentiation and substitution.

27 Hard

27 of 51

Analyzing +4 | -1

\int_0^1 \alpha e^{\beta x^2} \sin (x+k) d x=0

for some

\alpha, \beta \in R

\alpha \neq 0

, then the value of

k

can belong to the interval

View answer and explanation

Correct option: C

\left[\frac{3 \pi}{4}, \frac{5 \pi}{6}\right]

Option C ([3π/4, 5π/6]) is correct because the sine function must change sign within the interval [0, 1] for the integral to be zero.

28 Hard

28 of 51

Application +4 | -1

\int_{\pi / 2}^x \sqrt{\left(3-2 \sin ^2 t\right)} d t+\int_0^y \cos t d t=0

, then

\left(\frac{d y}{d x}\right)

x=\pi

and

y=\pi

View answer and explanation

Correct option: A

\sqrt{3}

Option A is correct because differentiating both sides of the given equation and then substituting the given values of x and y leads to the correct derivative value.

29 Hard

29 of 51

Analyzing +4 | -1

Let

f(x)=x+\frac{a}{\pi^{2}-4} \sin x+\frac{b}{\pi^{2}-4} \cos x, x \in \mathbb{R}

be a function which satisfies

f(x)=x+\int_{0}^{\pi / 2} \sin (x+y) f(y) d y

. Then

(a+b)

is equal to

View answer and explanation

Correct option: C

-2 \pi(\pi+2)

Option C (

-2 \pi(\pi+2)

) is correct because by substituting the given function into the integral equation and solving for a and b, we find that

a + b = -2\pi(\pi + 2)

30 Hard

30 of 51

Application +4 | -1

\{x\}

denotes the fractional part of

x

, then

\int\limits_{0}^x\left(\{x\}-\frac{1}{2}\right) d x

is equal to

View answer and explanation

Correct option: B

\frac{1}{2}\{x\}(\{x\}-1)

Option B (

\frac{1}{2}\{x\}(\{x\}-1)

) is correct because it represents the general form of the integral of the fractional part of x minus 1/2 from 0 to x.

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Key Takeaways

✓ This set contains 51 single-correct-answer MCQs on DEFINITE INTEGRATION.
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