Let a be a positive real number such that $\int_{\rm{0}}^{\rm{a}} {{{\rm{e}}^{x - \left[ x \right]}}dx} {\rm{ = 10e - 9}}\,$, where [x] is the greatest integer less than or equal to x. Then, a is equal to

${\rm{10 - lo}}{{\rm{g}}_e}\left( {1 + e} \right)$

${\rm{10 + lo}}{{\rm{g}}_e}\left( {1 + e} \right)$

$\mathop \smallint \nolimits_2^3 \,\,\frac{{dx}}{{{x^2} - x}}$ is equal to

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

$\log \,\,\left( {\frac{1}{4}} \right)$

$\log \,\,\left( {\frac{8}{3}} \right)$

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JEE • MATHEMATICS • DEFINITE INTEGRATION

Definite integration practice · Mixed difficulty

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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 811-840 of 995 questions

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811 Easy

811 of 995

Application +4 | -1

k=\int_0^1 \frac{e^t}{1+t} d t

, then

\int_0^1 e^t \log _e(1+t) d t

is equal to

View answer and explanation

Correct option: C

e \ln 2-k

The correct answer is C (

e \ln 2-k

) because after applying integration by parts, the integral is expressed in terms of k, resulting in

e \ln 2 - k

812 Easy

812 of 995

Analyzing +4 | -1

The value of

\int \limits_{\pi /4}^{3\pi /4} \,\,\frac{x}{{1\,\, + \,\,\sin \,\,x}}\,\,dx

is equal to

View answer and explanation

Correct option: A

\left( {\sqrt 2 - 1} \right)\pi

Option A ((

\sqrt 2 - 1)\pi

) is correct because the integral can be solved by using the property

\int_a^b f(x) dx = \int_a^b f(a+b-x) dx

and then simplifying the resulting expression.

813 Easy

813 of 995

Application +4 | -1

Let a be a positive real number such that

\int_\mathrm{0}^\mathrm{a} {{\mathrm{e}^{x - \left[ x \right]}}dx} \mathrm{ = 10e - 9}\,

, where [x] is the greatest integer less than or equal to x. Then, a is equal to

View answer and explanation

Correct option: B

\mathrm{10 + lo}{\mathrm{g}_e}2

Option B is correct because the integral can be solved by splitting it into intervals of length 1, utilizing the property of the greatest integer function, and then solving for 'a'.

814 Easy

814 of 995

Application +4 | -1

Solve

\int\limits_{0}^{\pi / 2} \frac{d x}{\left(a^2 \cos ^2 x+b^2 \sin ^2 x\right)^2}

View answer and explanation

Correct option: B

\frac{\pi}{4}\left(\frac{a^2+b^2}{a^3 b^3}\right)

Option B (

\frac{\pi}{4}\left(\frac{a^2+b^2}{a^3 b^3}\right)

) is correct because the integral can be solved using trigonometric substitution and partial fractions.

815 Easy

815 of 995

Application +4 | -1

\int \nolimits_2^3 \,\,\frac{{dx}}{{{x^2} - x}}

is equal to

View answer and explanation

Correct option: C

\log \,\,\left( {\frac{4}{3}} \right)

Option C is correct because the integral of 1/(x² - x) from 2 to 3 is solved using partial fraction decomposition, resulting in log(4/3).

816 Easy

816 of 995

Application +4 | -1

\lim \limits_{\mathrm{n} \rightarrow \infty} \frac{2^{\mathrm{k}}+4^{\mathrm{k}}+6^{\mathrm{k}}+\ldots+(2 \mathrm{n})^{\mathrm{k}}}{\mathrm{n}^{\mathrm{k}+1}}, \mathrm{k} \neq-1

, is equal to

View answer and explanation

Correct option: B

\frac{2^{\mathrm{k}}}{\mathrm{k}+1}

Option B (

\frac{2^{\mathrm{k}}}{\mathrm{k}+1}

) is correct because the limit can be expressed as a Riemann sum and then evaluated as a definite integral.

817 Hard

817 of 995

Understanding +4 | -1

Let

f: R \rightarrow R

be defined as

f(x)=\int\limits_{0}^1 \frac{x^2+t^2}{2-t} d t

. Then the curve

y=f(x)

is -

View answer and explanation

Correct option: D

A parabola

Option D (A parabola) is correct because after evaluating the integral, the resulting equation is a quadratic in

x

, which represents a parabola.

818 Medium

818 of 995

Analyzing +4 | -1

The value of

\int\limits_{{-\pi}}^{3 \pi} \log (\sec \theta-\tan \theta) d \theta

View answer and explanation

Correct option: B

Option B (0) is correct because the integral of

\log(\sec \theta - \tan \theta)

over the given interval is zero due to symmetry and properties of logarithms.

819 Easy

819 of 995

Application +4 | -1

The value of

\int \limits_0^1 \,\,\frac{{{2^{2x + 1}}\,\, - \,\,{5^{2x - 1}}}}{{{{10}^x}}}\,\,dx

, is

View answer and explanation

Correct option: B

- \frac{3}{5}\,\,\left\{ {\frac{2}{{{{\log }_e}\,\,\left( {\frac{2}{5}} \right)}}\,\, + \,\,\frac{1}{{2\,\,{{\log }_e}\,\,\left( {\frac{5}{2}} \right)}}} \right\}

Option B is correct because after simplifying the integrand and evaluating the definite integral, the result matches the given expression.

820 Easy

820 of 995

Application +4 | -1

Solution of the equation

\int\limits_{{\log _e 2}}^x \frac{d t}{\sqrt{e^t-1}}=\frac{\pi}{6}

View answer and explanation

Correct option: D

\log _e 4

Option D (log_e 4) is correct because after evaluating the integral and solving for x, the result is x = log_e 4.

821 Medium

821 of 995

Application +4 | -1

\int\limits_{{-2}}^2|x(x-1)| d x=

View answer and explanation

Correct option: A

\frac{17}{3}

Option A (

\frac{17}{3}

) is correct because after splitting the integral based on the absolute value and evaluating, the result is

\frac{17}{3}

822 Medium

822 of 995

Application +4 | -1

Let

f(x)

be a continuous function such that the area bounded by the curve

y=f(x)

, the

x

-axis, and the lines

x=0

and

x=a

1+\frac{a^2}{2} \sin a.

Then,

View answer and explanation

Correct option: C

f(a)=a \sin a+\frac{1}{2} a^2 \cos a

Option C (

f(a)=a \sin a+\frac{1}{2} a^2 \cos a

) is correct because it is obtained by differentiating the given integral equation using the Fundamental Theorem of Calculus and the product rule.

823 Medium

823 of 995

Application +4 | -1

A function

f(x)

safisfying

f(x)=\int\limits_{0}^x e^t f(t) d t+e^x

View answer and explanation

Correct option: A

2 e^{e^x-1}-1

Option A (

2 e^{e^x-1}-1

) is correct because it satisfies the given integral equation after differentiation and integration.

824 Easy

824 of 995

Application +4 | -1

The value of

\int\limits_{0}^{100}\left[\tan ^{-1} x\right] d x

, is

View answer and explanation

Correct option: C

100-\tan 1

Option C (

100 - \tan 1

) is correct because the integral simplifies to the length of the interval where the greatest integer of

\tan^{-1} x

is 1.

825 Easy

825 of 995

Application +4 | -1

I\left( {m,\;\,n} \right) = \int \nolimits_0^1 \,\,{t^m}{\left( {1 + \,\,t} \right)^n}\;\,dt

, then the expression for

I\left( {m,\;\,n} \right)

in terms of

I\left( {m + 1,\,\,\;n - 1} \right)

View answer and explanation

Correct option: A

\frac{{{2^n}}}{{m + 1}}\,\, - \,\,\frac{n}{{m + 1}}\,\;I\left( {m + 1,\;\,n - 1} \right)

Option A (

\frac{{{2^n}}}{{m + 1}}\, - \,\frac{n}{{m + 1}}\,\;I\left( {m + 1,\;\,n - 1} \right)

) is correct because it is derived by integration by parts.

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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

826 Medium

826 of 995

Understanding +4 | -1

A_n=\int\limits_{0}^{\pi / 2} \frac{\sin (2 n-1) x}{\sin x} d x

then value of

\frac{A_{1000}}{A_{10}}

is -

View answer and explanation

Correct option: A

Option D (Never less than

n

) is correct because, by the AM-GM inequality, the sum of positive numbers whose product is 1 is always greater than or equal to the number of terms.

827 Easy

827 of 995

Application +4 | -1

\lim\limits_{n \to \infty } \,\,\sum \nolimits_{r = 1}^n \,\,\frac{1}{n}\,\,{e^{r/n}}\,\,\mathrm{is}

View answer and explanation

Correct option: B

e\,\, - \,\,1

Option B (

e\,\, - \,\,1

) is correct because the limit of the sum can be expressed as a definite integral and then evaluated.

828 Easy

828 of 995

Understanding +4 | -1

I_1=\int\limits_{x}^1 \frac{1}{1+t^2} d t

and

I_2=\int\limits_{1}^1 \frac{1}{1+t^2} d t

for

x>0

, then

View answer and explanation

Correct option: A

I_1=I_2

Option A is correct because by substituting t = 1/u in I1, we can show that I1 is equal to I2.

829 Hard

829 of 995

Application +4 | -1

The value of

\frac{\pi^2}{\ln 3} \int_{7 / 6}^{5 / 6} \sec (\pi x) d x

, is

View answer and explanation

Correct option: C

\pi

The correct answer is C (

\pi

) because after evaluating the definite integral of the secant function and simplifying, the result is

\pi

830 Hard

830 of 995

Application +4 | -1

f(x)=e^{g(x)}

and

g(x)=\int_2^x \frac{t d t}{1+t^4}

, then

f^{\prime}(2)

is equal to

View answer and explanation

Correct option: A

2 / 17

Option A is correct because using the chain rule and the fundamental theorem of calculus, we find

f'(2) = 2/17

831 Easy

831 of 995

Application +4 | -1

Let [.] denote the greatest integer function, then the value of

\int\limits_{0}^{1.5} x\left[x^2\right] d x

, is

View answer and explanation

Correct option: B

\frac{3}{4}

Option B (

\frac{3}{4}

) is correct because the integral is evaluated by breaking it into intervals where the greatest integer function has a constant value.

832 Easy

832 of 995

Application +4 | -1

The value of

\int \nolimits_{ - 1}^{15} \,\,\mathrm{sgn}\,\,\left( {\left\{ x \right\}} \right)\,\,dx

, where

\left\{ {\, \cdot \,} \right\}

denotes the fractional part function, is

View answer and explanation

Correct option: B

\mathrm{16}

Option B (16) is correct because the integral of the sign of the fractional part function over the given interval simplifies to 16.

833 Medium

833 of 995

Analyzing +4 | -1

Let

[t]

denote the greatest integer less than or equal to t. Then the value of the integral

\int_{-3}^{101}\left([\sin (\pi x)]+e^{[\cos (2 \pi x)]}\right) d x

is equal to

View answer and explanation

Correct option: B

\frac{52}{e}

Option B is correct. The integral is evaluated by using the periodicity of the functions and simplifying the integral.

834 Medium

834 of 995

Application +4 | -1

If a function

f:[0,27] \rightarrow \mathbf{R}

is differentiable then for some

0<\alpha<\beta<3, \int\limits_{0}^{27} f(x) \mathrm{d} x

is equal to

View answer and explanation

Correct option: C

3\left[\alpha^2 f\left(\alpha^3\right)+\frac{1}{2} \beta^2 f\left(\beta^3\right)\right]

Option C is correct because it correctly applies the Mean Value Theorem for integrals and derivatives to the given function and interval.

835 Medium

835 of 995

Analyzing +4 | -1

Let m be any integer. Then, the integral

\int \limits_0^\pi \,\,\frac{{\sin 2\,\,m\mathrm{ }\,x}}{{\sin \,\,x}}\,\,dx

equals

View answer and explanation

Correct option: A

\mathrm{0}

Option A (0) is correct because the integral evaluates to zero due to the properties of trigonometric functions and the given limits.

836 Medium

836 of 995

Application +4 | -1

A minimum value of

\int \nolimits_0^x t{e^{{t^2}}}dt

View answer and explanation

Correct option: A

Option A (0) is correct because the minimum value of the integral is found by finding the critical points and checking the second derivative.

837 Medium

837 of 995

Application +4 | -1

The integral

\int\limits_ {\frac{\pi}{4}}^{\frac{3 \pi}{4}} \frac{\mathrm{dx}}{1+\cos \mathrm{x}}

is equal to:

View answer and explanation

Correct option: C

Option C (2) is correct because it is the value obtained after evaluating the definite integral of the given trigonometric function using appropriate identities and limits.

838 Easy

838 of 995

Application +4 | -1

\int\limits_{0}^{\infty} \frac{d x}{\left(x^2+4\right)\left(x^2+9\right)}=k \pi

then the value of

k

View answer and explanation

Correct option: A

1 / 60

Option A (

1 / 60

) is correct because the integral is solved using partial fractions and the arctangent integral.

839 Easy

839 of 995

Application +4 | -1

The numbers

A, B

and

C

such that a function of the form

f(x)=A x^2+B x+C

satisfies the conditions

f^{\prime}(1)=8, f(2)+f^{\prime \prime}(2)=33

and

\int\limits_{0}^1 f(x) \mathrm{d} x=7 / 3

, are

View answer and explanation

Correct option: B

A=7, B=-6, C=3

Option B (

A=7, B=-6, C=3

) is correct because these values satisfy the given conditions

f'(1)=8

f(2)+f''(2)=33

, and

\int_0^1 f(x) dx = 7/3

840 Medium

840 of 995

Analyzing +4 | -1

The integral

\int\limits_0^1 {\frac{1}{{{7^{\left[ {\frac{1}{x}} \right]}}}}} dx,

where

\left[ \cdot \right]

denotes the greatest integer function, is equal to

View answer and explanation

Correct option: A

1 + 6{\log _e}\left( {\frac{6}{7}} \right)

The correct answer is A (

1 + 6{\log _e}(\frac{6}{7})

) because it is the result of evaluating the given definite integral using substitution and properties of logarithms.

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