If $f(x)=a+b x+c x^2$ where $a, b, c \in R$, then $\int_0^1 f(x) d x$ is equal to

$\frac{1}{3}\left\{f(1)+f(0)+2 f\left(\frac{1}{2}\right)\right\}$

If $f\left( {a + x} \right) = f\left( x \right)$, then $\mathop \smallint \limits_0^{na} \,f\,\,\left( x \right)\,\,dx$ is equal to $\left( {n \in N} \right)$

$n\,\mathop \smallint \limits_0^a \,f\left( x \right)\,\,dx$

$\left( {n - 1} \right)\,\,\mathop \smallint \limits_0^a f\left( x \right)\,\,dx$

$\mathop \smallint \limits_0^{\left( {n - 1} \right)a} \,f\left( x \right)\,\,dx$

The value of $\mathop \smallint \limits_0^{16\;\pi /3} \,\,\left| {\,\sin \,\,x\,} \right|\,\,dx$, is

${\rm{21}}\,{\rm{/}}\,{\rm{2}}$

$\mathop \smallint \limits_{1/2}^2 \,\,\left| {\,{{\log }_{10}}\,\,x\,} \right|dx$ is equals to

${\log _{10}}\,\left( {2/e} \right)$

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JEE • MATHEMATICS • DEFINITE INTEGRATION

Definite integration practice · Easy

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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 151-180 of 791 questions

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151 Easy

151 of 791

Application +4 | -1

The value of

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

is:

View answer and explanation

Correct option: A

\pi^2

Option A (

\pi^2

) is correct because the integral simplifies to

\int_0^{2 \pi} x dx

, which evaluates to

\pi^2

152 Easy

152 of 791

Application +4 | -1

Let

f(x)=\int\limits_{0}^x \sqrt{6-u^2} d u

. Then the real roots of the equation

x^2-f^{\prime}(x)=0

are

View answer and explanation

Correct option: C

x= \pm \sqrt{2}

Option C is correct because the real roots are found by solving the equation after finding the derivative of the integral and setting it equal to x squared.

153 Easy

153 of 791

Application +4 | -1

Let

f(x)=x^2, g(x)=\cos x

and

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

. Area bounded by

y=h(x)

and

y

-axis between

x=x_1

and

x=x_2

, where

x_1

and

x_2

are roots of the equation

18 x^2-9 \pi x+\pi^2=0

, is equal to

View answer and explanation

Correct option: A

\frac{\pi}{12}

sq. units

Option A is correct because it represents the area calculated by integrating cos²(x) from π/6 to π/3.

154 Easy

154 of 791

Analyzing +4 | -1

\lim \limits_{{n \rightarrow \infty}} \sum\limits_{{r=2 n+1}}^{3 n} \frac{n}{r^2-n^2}

is equal to

View answer and explanation

Correct option: B

\ln \sqrt{\frac{3}{2}}

Option B (

\ln \sqrt{\frac{3}{2}}

) is correct because the limit of the sum can be expressed as a definite integral, which evaluates to

\ln \sqrt{\frac{3}{2}}

155 Easy

155 of 791

Analyzing +4 | -1

The integral

16 \int\limits_{{1}}^{2} \frac{d x}{x^{3}\left(x^{2}+2\right)}

is equal to

View answer and explanation

Correct option: B

\frac{11}{6}-\log _{e} 4

Option B (

\frac{11}{6}-\log _{e} 4

) is correct because after evaluating the integral, the result matches this option.

156 Easy

156 of 791

Understanding +4 | -1

f(x)=a+b x+c x^2

where

a, b, c \in R

, then

\int_0^1 f(x) d x

is equal to

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because by evaluating the integral

\int_0^1 f(x) dx

and comparing it with the given options, we find that the correct expression is

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

157 Easy

157 of 791

Application +4 | -1

\int \nolimits_{ - 2}^{10} \,\mathrm{sgn}\,(x - \,\,\left[ x \right]\,dx

Equals, where

\left[ {\,\mathrm{.}\,} \right]\,

denotes greatest integer function

View answer and explanation

Correct option: C

\mathrm{8}

Option C (8) is correct because the integral of the sign function simplifies to the difference between the lengths of the positive and negative intervals.

158 Easy

158 of 791

Application +4 | -1

\int\limits_{0}^1 \frac{\sin t}{1+t} d t=\alpha

, then the value of the integral

\int\limits_{{4 \pi-2}}^{4 \pi} \frac{\sin \frac{t}{2}}{4 \pi+2-t} d t

in terms of

\alpha

is given by

View answer and explanation

Correct option: D

-\alpha

Option D (

-\alpha

) is correct because after substitution and simplification, the integral becomes the negative of the given integral.

159 Easy

159 of 791

Application +4 | -1

The value of

\int\limits_{{-\pi}}^\pi \frac{\cos ^2 x}{1+a^x} d x, a > 0

, is

View answer and explanation

Correct option: C

\pi / 2

Option C (

\pi / 2

) is correct because by using the property

\int_0^a f(x) dx = \int_0^a {f(x) + f(-x)} dx

, the integral simplifies to

\int_0^\pi \cos^2 x dx

, which evaluates to

\pi/2

160 Easy

160 of 791

Analyzing +4 | -1

\int \nolimits_0^2 \,\,\sqrt {\frac{{2\,\, + \,\,x}}{{2\,\, - \,\,x}}} \,\,dx

is equal to

View answer and explanation

Correct option: A

\pi + 2

Option A is correct because after simplifying and using trigonometric substitution, the integral evaluates to π + 2.

161 Easy

161 of 791

Understanding +4 | -1

Let

f\;:R \to R,\,\;g\;:R \to R

be continuous functions. Then the value of the integral

\int \limits_{ - \pi /2}^{\pi /2} \,\,\left[ {\,f\left( x \right)\,\, + \,\,f\left( { - x} \right)} \right]\,\,\left[ {\,\,g\,\,\left( x \right) - \,\,g\left( {\, - \,x} \right)} \right]\,\,dx

View answer and explanation

Correct option: D

\mathrm{0}

Option D (0) is correct because the integrand is an odd function integrated over a symmetric interval.

162 Easy

162 of 791

Application +4 | -1

\{x\}

denotes the fractional part of

x

, then the value of the integral

\int\limits_{2}^4\{2 x\} d x

View answer and explanation

Correct option: B

Option B (1) is correct because the integral simplifies due to the periodic nature of the fractional part function.

163 Easy

163 of 791

Application +4 | -1

\int\limits_{\sqrt{2}}^x \frac{d t}{t \sqrt{t^2-1}}=\frac{\pi}{12}

, then

x

equals

View answer and explanation

Correct option: C

Option C (2) is correct because after evaluating the integral and solving for x, the result is x=2.

164 Easy

164 of 791

Analyzing +4 | -1

The equation

\int \limits_{ - \pi /4}^{\pi /4} \,\,\left\{ {a\,\,\left| {\,\sin \,\,x\,} \right|\,\, + \,\,\frac{{b\,\,\sin \,\,x}}{{1\, + \,\,\cos \,\,x}}\,\, + \,\,c} \right\}dx = 0

, where

a,\;\,b,\;\,c

are constants, gives a relation between

View answer and explanation

Correct option: B

a

and

c

Option B (a and c) is correct because the integral of the term with 'b' vanishes due to it being an odd function, leaving a relation only between 'a' and 'c'.

165 Easy

165 of 791

Application +4 | -1

\int\limits_{0}^{\pi / 2} \sin 2 x \log \tan x d x

is equal to

View answer and explanation

Correct option: C

Option C (0) is correct because the integral can be solved using the property

\int_0^a f(x) dx = \int_0^a f(a-x) dx

and showing that the integral is equal to its negative.

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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

166 Easy

166 of 791

Application +4 | -1

The value of

\int\limits_{0}^2 x d([x]-x)

, is

View answer and explanation

Correct option: B

Option B (1) is correct because the integral is evaluated using integration by parts and properties of the greatest integer function.

167 Easy

167 of 791

Application +4 | -1

Area bounded by curve

y=k \sin x

between

x=\pi

and

x=2 \pi

, is

View answer and explanation

Correct option: A

2 k

sq. unit

Option A (

2k

sq. unit) is correct because the area is calculated by integrating

|k \sin x|

from

\pi

2\pi

, which simplifies to

2k

168 Easy

168 of 791

Application +4 | -1

The value of

\int \nolimits_{ - 2}^3 \,\,\left| {\,1 - {x^2}\,} \right|\,\,dx\,\,\mathrm{is}

View answer and explanation

Correct option: A

\frac{{28}}{3}

Option A (

\frac{{28}}{3}

) is correct because it is the result of correctly evaluating the definite integral of the absolute value function over the given interval.

169 Easy

169 of 791

Understanding +4 | -1

f\left( {a + x} \right) = f\left( x \right)

, then

\int \limits_0^{na} \,f\,\,\left( x \right)\,\,dx

is equal to

\left( {n \in N} \right)

View answer and explanation

Correct option: B

n\,\int \limits_0^a \,f\left( x \right)\,\,dx

Option B (

n\int \limits_0^a \\,f\left( x \right)\\,,dx

) is correct because the integral over

na

can be split into

n

integrals over the interval

[0, a]

due to the periodicity of

f(x)

170 Easy

170 of 791

Application +4 | -1

The value of

\int \nolimits_{ - 2}^4 \,\,\left| {x + 1} \right|\,\,dx

is equal to

View answer and explanation

Correct option: C

\mathrm{13}

Option C is correct because the integral of |x+1| from -2 to 4 is calculated by splitting the integral at x = -1, resulting in a value of 13.

171 Easy

171 of 791

Application +4 | -1

The value of

\int \nolimits_0^\infty \,\,\frac{{dx}}{{\left( {{x^2} + \,\,4} \right)\,\,\left( {{x^2} + \,\,9} \right)}}

View answer and explanation

Correct option: A

\frac{\pi }{{60}}

Option A (

\frac{\pi}{{60}}

) is correct because after using partial fractions and integrating, the result is

\frac{\pi}{{60}}

172 Easy

172 of 791

Understanding +4 | -1

\int\limits_{{-a}}^a \sin x f(\cos x) d x=

View answer and explanation

Correct option: B

Option B is correct because sin(x) is an odd function and f(cos(x)) is an even function, making the product an odd function, and the integral of an odd function over a symmetric interval is zero.

173 Easy

173 of 791

Analyzing +4 | -1

The value of

\int\limits_0^{2\pi } {\left[ {\sin 2x(1 + \cos 3x)} \right]dx,}

where

[t]

denotes the greatest integer function, is

View answer and explanation

Correct option: A

- \pi

Option A (

\ - \pi

) is correct because the integral simplifies to

\int_0^{2\pi} [\sin 2x (1 + \cos 3x)] dx = -\pi

after using the property

[x] + [-x] = -1

for non-integer

x

174 Easy

174 of 791

Application +4 | -1

The function

f(x),

that satisfies the condition

f(x) = x + \int\limits_0^{\pi /2} {\sin x.\cos y\,\,f(y)dy,}

View answer and explanation

Correct option: D

x + (\pi - 2)\sin x

Option D (

x + (\pi - 2)\sin x

) is the correct function because it satisfies the given integral equation.

175 Easy

175 of 791

Application +4 | -1

f(x)=\left\{\begin{array}{ll}x^2, & 0 \leq x \leq 1 \\ \sqrt{x}, & 1 \leq x \leq 2\end{array}\right.

then

\int\limits_{0}^2 f(x) d x=

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the calculated integral value does not match any of the given options.

176 Easy

176 of 791

Analyzing +4 | -1

A=\sum\limits_{{r=1}}^{\infty}\left(\int\limits_{0}^1 \frac{x^{r-1}}{r} d x\right), B=\sum\limits_{{r=1}}^{\infty}\left(\int\limits_{0}^1 \frac{x^{2 r-2}}{2 r-1} d x\right)

then

\frac{3 A+B}{B}

View answer and explanation

Correct option: C

Option C (5) is correct because it is the simplified ratio of the sums A and B after evaluating the integrals and recognizing the resulting series.

177 Easy

177 of 791

Analyzing +4 | -1

\int \nolimits_{ - 1}^1 \,\,\left( {{e^x}^3 + \,\,{e^{ - x}}^3} \right)\left( {{e^x} - {e^{ - x}}} \right)dx\,\,\mathrm{is\;equal\;to}

View answer and explanation

Correct option: D

\mathrm{0}

Option D (0) is correct because the integrand is an odd function, and the integral of an odd function over a symmetric interval is zero.

178 Easy

178 of 791

Application +4 | -1

The value of the sum

\sum\limits_{{n=1}}^{100} \int\limits_{{n-1}}^n e^{x-[x]} d x

, where

[\cdot]

denotes the greatest integer function, is

View answer and explanation

Correct option: B

100(e-1)

Option B (

100(e-1)

) is correct because the sum of integrals can be combined into a single integral, which simplifies due to the periodic nature of the function.

179 Easy

179 of 791

Application +4 | -1

The value of

\int \limits_0^{16\;\pi /3} \,\,\left| {\,\sin \,\,x\,} \right|\,\,dx

, is

View answer and explanation

Correct option: B

\mathrm{21}\,\mathrm{/}\,\mathrm{2}

Option B (

\mathrm{21}\,\mathrm{/}\,\mathrm{2}

) is correct because it involves integrating the absolute value of sine over multiple periods.

180 Easy

180 of 791

Application +4 | -1

\int \limits_{1/2}^2 \,\,\left| {\,{{\log }_{10}}\,\,x\,} \right|dx

is equals to

View answer and explanation

Correct option: B

\frac{1}{2}\,\,{\log _{10}}\,\,\left( {8/e} \right)

Option B (

\frac{1}{2}\,{\log _{10}}\,\,\left( {8/e} \right)

) is correct because it is the result of evaluating the definite integral after splitting it based on the absolute value of the logarithm.

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