The value of the integral $\mathop \smallint \limits_{ - \pi /3}^{\pi /3} \,\,\frac{{x\,\,\sin \,\,x}}{{{{\cos }^2}\,\,x}}\,\,dx$, is

$\left( {\pi /3\,\, - \,\,\log \,\,\tan \,\,3\pi /2} \right)$

$3\,\,\left( {\pi /2\,\, - \,\,\log \,\,\sin \,\,\pi /12} \right)$

The value of $\int_\limits 0^4 3^{\sqrt{2 x+1}} \mathrm{~d} x$ is

$\frac{6}{\log 3}\left(13-\frac{4}{\log 3}\right)$

$\frac{6}{\log 3}\left(13-\frac{5}{\log 3}\right)$

$\mathop \smallint \limits_0^{\pi /2} \,\,\left( {x\,\,\sqrt {\tan \,\,x} \,\, + \,\,\sqrt {\cot \,\,x} } \right)\,\,dx$ equals

$\frac{{{\pi ^2}}}{{2\,\sqrt 3 }}$

The value of $\mathop \smallint \limits_0^{\pi /2} \frac{1}{{9\,\,\cos \,\,x + 12\,\,\sin \,\,x}}\,\,dx$, is

$\frac{1}{{15}}\,\,{\log _{10}}\,\,6$

$\log \,\,\left( {\frac{6}{{15}}} \right)$

$\log \,\,\left( {\frac{{15}}{6}} \right)$

The value of the integral $\mathop \smallint \limits_0^1 \,\,x{\left( {1 - x} \right)^n}\,\,dx$, is

$\frac{1}{{n + 1}}\,\, + \,\,\frac{1}{{n + 2}}$

$\frac{1}{{n + 2}}\,\, - \,\,\frac{1}{{n + 1}}$

$2\,\,\left( {\frac{1}{{n + 1}}\,\, - \,\,\frac{1}{{n + 2}}} \right)$

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JEE • MATHEMATICS • DEFINITE INTEGRATION

Definite integration practice · Medium

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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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271 Medium

271 of 596

Application +4 | -1

\int\limits_{0}^{100} f(x) d x=a

, then

\sum\limits_{{r=1}}^{100} \int\limits_{0}^1(f(r-1+x) d x)=

View answer and explanation

Correct option: B

a

Option B (

a

) is correct because the sum of the integrals can be combined into a single integral from 0 to 100, which is given as

a

272 Medium

272 of 596

Analyzing +4 | -1

\int \nolimits_0^{\pi /2} \,\,\log \,\,\sin \,\,x\,\,dx

is equal to

View answer and explanation

Correct option: C

- \,\,\frac{\pi }{2}\,\,\log \,\,2

Option C

\left( { - \,\,\frac{\pi }{{2}}\,\log \,2} \right)

is correct because by using the property

\int \nolimits_0^{a} f(x) dx = \int \nolimits_0^{a} f(a-x) dx

and trigonometric identities, the integral evaluates to

-\frac{\pi}{2}\log 2

273 Medium

273 of 596

Application +4 | -1

The value of

\int\limits_{{-1}}^1(x-[x]) d x

(where

[

] denotes greatest

integer function) is

View answer and explanation

Correct option: B

Option B (1) is correct because the integral evaluates to 1 after splitting the integral and using the properties of the greatest integer function.

274 Medium

274 of 596

Application +4 | -1

If the integral

\int \frac{\cos 8 x+1}{\cot 2 x-\tan 2 x} d x=A \cos 8 x+k

where

k

is an arbitrary constant, then

A

is equal to:

View answer and explanation

Correct option: A

-\frac{1}{16}

Option A (-1/16) is correct because the integral simplifies to a cosine function with a coefficient of -1/16.

275 Medium

275 of 596

Application +4 | -1

If the

\int \frac{5 \tan x}{\tan x-2} d x=x+a \ln |\sin x-2 \cos x|+k

, then

a

is equal to:

View answer and explanation

Correct option: D

Option D (2) is correct because after simplifying the integral, the coefficient of the natural logarithm term is 2.

276 Medium

276 of 596

Application +4 | -1

The value of the integral

\int \limits_{ - \pi /3}^{\pi /3} \,\,\frac{{x\,\,\sin \,\,x}}{{{{\cos }^2}\,\,x}}\,\,dx

, is

View answer and explanation

Correct option: B

2\,\,\left( {2\pi /3\,\, - \,\,\log \,\,\tan \,\,5\pi /12} \right)

Option B (

2\,\left( {2\pi /3\, - \,\log \,\tan \,5\pi /12} \right)

) is correct because it is the result of evaluating the definite integral using integration by parts and properties of even functions.

277 Medium

277 of 596

Analyzing +4 | -1

\int\limits_{0}^{\pi / 2} f(\sin 2 x) \sin x d x=k \int\limits_{0}^{\pi / 4} f(\cos 2 x) \cos x d x

where

k

equals

View answer and explanation

Correct option: C

\sqrt{2}

Option C (

\sqrt{{2}}

) is correct because after applying the properties of definite integrals and trigonometric identities, the value of k is found to be

\sqrt{{2}}

278 Medium

278 of 596

Application +4 | -1

\int_0^{16 n^2 / \pi} \cos \frac{\pi}{2}\left[\frac{x \pi}{n}\right] d x

is equal to

View answer and explanation

Correct option: A

Option A is correct because by substituting and using the properties of the greatest integer function and cosine, the integral evaluates to 0.

279 Medium

279 of 596

Application +4 | -1

f\left( \pi \right) = 2

and

\int \nolimits_0^\pi \,\,\left[ {f\left( x \right)\,\, + \,\,f''\left( x \right)} \right]\,\,\sin \,\,x\,\,dx = 5,

, then

f\,\,\left( 0 \right)

is equal to, (it is given that

f\left( x \right)

is continuous in

\left[ {0,\,\,\pi } \right]\,

)

View answer and explanation

Correct option: B

Option B (3) is correct because integration by parts is applied to simplify the integral, leading to the equation

f(\pi) + f(0) = 5

, and given

f(\pi) = 2

, we find

f(0) = 3

280 Medium

280 of 596

Analyzing +4 | -1

\int_0^{\infty} f\left(x+\frac{1}{x}\right) \cdot \frac{\ln x}{x} d x

is equal to

View answer and explanation

Correct option: A

Option A (0) is correct because the integral evaluates to zero due to the odd symmetry of the function after substitution.

281 Medium

281 of 596

Application +4 | -1

The value of

\int \limits_0^1 \left( {1\,\, + \,\,{e^{ - {x^2}}}} \right)dx

, is

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the definite integral's value falls within a range that doesn't match any of the provided options. The integral cannot be expressed in a simple closed form.

282 Medium

282 of 596

Application +4 | -1

\int\limits_{0}^{\frac{\pi}{2}} \ln |\tan x+\cot x| d x=\lambda \ln 2

, then

\lambda

equals

View answer and explanation

Correct option: B

\pi

Option B (

\pi

) is correct because the integral simplifies to

\pi \ln 2

after using trigonometric identities and properties of logarithms, thus

\lambda = \pi

283 Medium

283 of 596

Application +4 | -1

The value of

\int\limits_{0}^{\pi / 2} \frac{\sin ^3 x}{\sin x+\cos x} d x

is:

View answer and explanation

Correct option: B

\frac{\pi-1}{4}

Option B (

\frac{{\pi-1}}{{4}}

) is correct because it is the result obtained after applying the property of definite integrals and simplifying the expression.

284 Medium

284 of 596

Application +4 | -1

The value of

\alpha

for which

4 \alpha \int\limits_{{-1}}^2 e^{-\alpha|x|} d x=5

, is :

View answer and explanation

Correct option: A

\log _e 2

Option A (

\log_e 2

) is correct because it is the value of

\alpha

that satisfies the given integral equation after solving for

\alpha

285 Medium

285 of 596

Application +4 | -1

The value of

\int\limits_{0}^4 3^{\sqrt{2 x+1}} \mathrm{~d} x

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the integral evaluates to a value not present in the given options.

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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

286 Medium

286 of 596

Analyzing +4 | -1

I = \int \limits_0^{1/2} \,\,\frac{1}{{\sqrt {1 - {x^{2n}}} }}\,\,dx

, then which one of the following is true?

View answer and explanation

Correct option: D

All of these

Option D (All of these) is correct because the integral I is bounded by

\frac{1}{2}

and

\frac{\pi}{6}

, and it is also greater than 0.

287 Medium

287 of 596

Application +4 | -1

The value of

\int \nolimits_0^\mathrm{\pi } \,\,|{\sin ^3}\,\,\mathrm{\theta }|\,\,d\mathrm{\theta }

View answer and explanation

Correct option: C

\mathrm{4}\,\mathrm{/}\,\mathrm{3}

Option C (

\frac{4}{3}

) is correct because the integral of |sin³θ| from 0 to π is equal to the integral of sin³θ from 0 to π, which evaluates to 4/3.

288 Medium

288 of 596

Application +4 | -1

\int \limits_0^{\pi /2} \,\,\left( {x\,\,\sqrt {\tan \,\,x} \,\, + \,\,\sqrt {\cot \,\,x} } \right)\,\,dx

equals

View answer and explanation

Correct option: C

\frac{{{\pi ^2}}}{{2\,\,\sqrt 2 }}

Option C (

\frac{{{\pi ^2}}}{{2\sqrt 2 }}

) is correct because it is the result of evaluating the definite integral using properties of definite integrals and trigonometric substitutions.

289 Medium

289 of 596

Application +4 | -1

Let

I_1=\int\limits_{a}^{\pi-a} x f(\sin x) d x, I_2=\int\limits_{a}^{\pi-a} f(\sin x) d x

, then

I_2

is equal to

View answer and explanation

Correct option: C

\frac{2}{\pi} I_1

Option C (

\frac{2}{\pi} I_1

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

I_1

and

I_2

, we can relate the two integrals and solve for

I_2

290 Medium

290 of 596

Application +4 | -1

\mathop {{\text{Lim}}}\limits_{n \to \infty } \sum\limits_{r = 1}^n {\frac{1}{n}} {e^{\frac{r}{n}}}\,\,{\text{is}}

View answer and explanation

Correct option: B

e-1

Option B (

e-1

) is correct because the limit of the Riemann sum is evaluated as a definite integral from 0 to 1 of

e^x

291 Medium

291 of 596

Application +4 | -1

The value of

\int \limits_0^{\pi /2} \frac{1}{{9\,\,\cos \,\,x + 12\,\,\sin \,\,x}}\,\,dx

, is

View answer and explanation

Correct option: B

\frac{1}{{15}}\,\,{\log _e}\,\,6

Option B (

\frac{{1}}{{15}}\,{\log _e}\,6

) is correct because it is the result of evaluating the definite integral using trigonometric substitution and logarithmic properties.

292 Medium

292 of 596

Application +4 | -1

Area enclosed by the curve

\mathrm{\pi }[4\left( {\mathrm{x} - \sqrt 2 {)^2} + {y^2}} \right] = 8

View answer and explanation

Correct option: D

\mathrm{4\;sq\;units\;}

Option D,

\mathrm{4\;sq\;units\;}

, is the correct area because the equation represents an ellipse, and its area is calculated using the formula

\pi ab

, where a and b are the semi-major and semi-minor axes.

293 Medium

293 of 596

Analyzing +4 | -1

\int\limits_{0}^x e^{z x} \cdot e^{-z^2} d z=f(x) \int\limits_{0}^x e^{-z^2 / 4} d z

then

\int e^x\left(\log _e(f(x))+\frac{x}{2}\right) d x=

View answer and explanation

Correct option: B

\frac{x^2 e^x}{4}+c

Option B (

\frac{{x^2 e^x}}{{4}}

+c) is correct because after solving the given integral equation for f(x), the resulting integral can be evaluated to get this answer.

294 Medium

294 of 596

Application +4 | -1

The value of the definite integral

\int\limits_{0}^{2 \pi} e^{\cos \theta} \cos \theta(\sin \theta) d \theta

View answer and explanation

Correct option: C

2 \pi

Option C (

2 \pi

) is correct because after expanding the exponential and integrating term by term, only the constant term contributes to the integral.

295 Medium

295 of 596

Analyzing +4 | -1

The value of the constant

a>0

such that

\int_0^a\left[\tan ^{-1} \sqrt{x}\right] d x=\int_0^a\left[\cot ^{-1} \sqrt{x}\right] d x

, where

[\cdot]

denotes the greatest integer function, is

View answer and explanation

Correct option: A

\frac{2(3+\cos 4)}{1-\cos 4}

Option A (

\frac{2(3+\cos 4)}{1-\cos 4}

) is correct because it is derived by solving the equation after simplifying the integrals using the greatest integer function.

296 Medium

296 of 596

Analyzing +4 | -1

Let

f

be a positive valued function. Let

I_1=\int\limits_{{1-k}}^k x f[x(1-x)] d x

and

I_2=\int\limits_{{1-k}}^k f[x(1-x)] d x

then

I_1 / I_2

View answer and explanation

Correct option: C

\frac{1}{2}

Option C (

\frac{{1}}{{2}}

) is correct because by applying the property of definite integrals,

I_1

can be expressed in terms of

I_2

, leading to the ratio

\frac{{I_1}}{{I_2}} = \frac{{1}}{{2}}

297 Medium

297 of 596

Analyzing +4 | -1

Let

f: R \rightarrow R

be a function which satisfies

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

. Then the value of

f(\ln 5)

, is

View answer and explanation

Correct option: B

Option B (0) is correct because the function satisfying the given condition is the zero function, making f(ln 5) equal to 0.

298 Medium

298 of 596

Analyzing +4 | -1

The ratio of the area of the region bounded by the curve

{a^4}{y^2} = (2a - x){x^5}

to the area of the
circle whose radius is

a,

is given by

View answer and explanation

Correct option: B

05:04

Option B (05:04) is correct because the ratio of the area under the curve to the area of the circle is calculated to be 5/4.

299 Medium

299 of 596

Analyzing +4 | -1

The value of

\lim\limits _{x \rightarrow 0} \frac{1}{x^3} \int_0^1 \frac{t \ln (1+t)}{t^4+4} d t

, is

View answer and explanation

Correct option: B

1 / 12

Option B (

1/12

) is correct because applying L'Hopital's rule twice yields this limit.

300 Medium

300 of 596

Application +4 | -1

The value of the integral

\int \limits_0^1 \,\,x{\left( {1 - x} \right)^n}\,\,dx

, is

View answer and explanation

Correct option: B

\frac{1}{{\left( {n + 1} \right)\,\,\left( {n + 2} \right)}}

Option B (

\frac{1}{{\left( {n + 1} \right)\,\,\left( {n + 2} \right)}}

) is correct because after applying the property of definite integrals and simplifying, we arrive at this result.

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