The value of the integral $\mathop \smallint \limits_{ - 1/2}^{1/2} \,\,{\left\{ {{{\left( {\frac{{x + 1}}{{x - 1}}} \right)}^2} + \,\,{{\left( {\frac{{x - 1}}{{x + 1}}} \right)}^2} - \,\,2} \right\}^{1/2}}dx$, is

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

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

The value of $\mathop \smallint \limits_a^{a + \left( {\pi /2} \right)} \,\,\left( {{{\sin }^4}\,\,x\,\, + \,\,{{\cos }^4}\,\,x} \right)\,\,dx$, is

$a\,\,{\left( {\frac{\pi }{2}} \right)^2}$

$ f(x)=\int \frac{2 x}{\left(x^{2}+1\right)\left(x^{2}+3\right)} d x $ if $ f(3)=\frac{1}{2}[\ln 5-\ln 6] $, then $ f(4) $ is:

$ \frac{1}{2}[\ln 19-\ln 17] $

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JEE • MATHEMATICS • DEFINITE INTEGRATION

Definite integration practice · Medium

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361 Medium

361 of 596

Application +4 | -1

\int\limits_{0}^2 \frac{x^3}{\left(x^2+1\right)^{3 / 2}} d x

is equal to

View answer and explanation

Correct option: D

None of these

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

362 Medium

362 of 596

Application +4 | -1

\int\limits_{3}^{10}[\log [x]] d x

is equal to (where[.]represents the greatest integer function)

View answer and explanation

Correct option: A

Option A (9) is correct because the integral can be evaluated by splitting the interval into subintervals where the greatest integer function is constant.

363 Medium

363 of 596

Application +4 | -1

\int\limits_{{-2}}^3\left|1-x^2\right| d x=

View answer and explanation

Correct option: D

\frac{28}{3}

Option D is correct because by splitting the integral based on the absolute value function's behavior, we can evaluate each part and sum them up to get

\frac{28}{3}

364 Medium

364 of 596

Application +4 | -1

\int \nolimits_0^{\pi /4} \,\,(\cos \,\,x\,\, - \,\,\sin x) + \int \nolimits_{\pi /4}^{5\pi /4} (\sin \,\,x\,\, - \,\,\cos \,\,x)\,\,dx\,\, + \,\,\int \nolimits_{2\pi }^{\pi /4} \,\,(\cos \,\,x\,\, - \,\,\sin \,\,x)\,\,dx

is equal to

View answer and explanation

Correct option: D

4\,\sqrt 2 - 2

Option D (

4\sqrt 2 - 2

) is correct because after evaluating each integral and summing the results, the final value is

4\sqrt{2} - 2

365 Medium

365 of 596

Analyzing +4 | -1

\int \frac{d x}{\cos ^3 x \sqrt{2 \sin 2 x}}=(\tan x)^{\mathrm{A}}+\mathrm{C}(\tan x)^B+k

, where

k

is a constant of integration, then

\mathrm{A}+\mathrm{B}+\mathrm{C}

equals:

View answer and explanation

Correct option: A

\frac{16}{5}

Option A (

\frac{16}{5}

) is correct. By substituting tan x = t² and solving the integral, we find A, B, and C, and their sum is 16/5.

366 Medium

366 of 596

Analyzing +4 | -1

\int\limits_{0}^x f(\mathrm{t}) \mathrm{dt}=x^2+\int\limits_{x}^1 \mathrm{t}^2 f(\mathrm{t}) \mathrm{dt}

, then

f^{\prime}(1 / 2)

is:

View answer and explanation

Correct option: A

\frac{24}{25}

Option A (

\frac{24}{25}

) is correct because it is the value of

f'(1/2)

obtained after differentiating the given integral equation and solving for

f'(x)

367 Medium

367 of 596

Analyzing +4 | -1

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

View answer and explanation

Correct option: A

\frac{\pi}{2} \log \frac{1}{2}

The correct answer is A (

\frac{\pi}{2} \log \frac{1}{2}

) because by using the property of definite integrals and trigonometric identities, the integral simplifies to

\frac{\pi}{2} \log \frac{1}{2}

368 Medium

368 of 596

Analyzing +4 | -1

\int\limits_{e}^x t f(t) d t=\sin x-x \cos x-\frac{x^2}{2}

, for all

x \in R-\{0\}

, then the value of

f\left(\frac{\pi}{6}\right)

View answer and explanation

Correct option: D

-1 / 2

Option D (-1/2) is correct because it is the result of differentiating the given equation and substituting the value of x.

369 Medium

369 of 596

Application +4 | -1

The value of the integral

\int \limits_{ - 1/2}^{1/2} \,\,{\left\{ {{{\left( {\frac{{x + 1}}{{x - 1}}} \right)}^2} + \,\,{{\left( {\frac{{x - 1}}{{x + 1}}} \right)}^2} - \,\,2} \right\}^{1/2}}dx

, is

View answer and explanation

Correct option: C

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

Option C,

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

, is correct because it is the result of simplifying the integrand and evaluating the definite integral.

370 Medium

370 of 596

Application +4 | -1

\int\limits_{0}^{\pi / 2} \frac{d x}{a^2 \cos ^2 x+{b}^2 \sin ^2 x}=\frac{\pi}{16}

. Then minimum value of

a \cos x+b \sin x

View answer and explanation

Correct option: A

-4

Option A (-4) is correct because the minimum value of the expression is found by relating the integral to ab and minimizing

a \cos x + b \sin x

using

a=b

371 Medium

371 of 596

Application +4 | -1

[x]

denotes the greatest integer less than or equal to

x

then the value of

\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}}[[x]-\operatorname{Sin} x] d x=

View answer and explanation

Correct option: D

-\pi

Option D (

-\pi

) is correct because the integral simplifies to the integral of -1 over the given interval, which results in

-\pi

372 Medium

372 of 596

Application +4 | -1

The value of

\int\limits_{{0}}^{16 \pi / 3}|\sin x| d x

View answer and explanation

Correct option: C

21 / 2

Option C (21/2) is correct because the integral of the absolute value of sin(x) over the given interval is calculated by splitting the integral into intervals of length pi.

373 Medium

373 of 596

Analyzing +4 | -1

Let

f(x)=x^2+\int_0^x e^{-t} f(x-t) d t

and

f(0)=0

. Then

[f(3)]

where [.] represent greatest integer

View answer and explanation

Correct option: B

Option B (18) is correct because after solving the integral equation and substituting x=3, the greatest integer function evaluates to 18.

374 Medium

374 of 596

Application +4 | -1

The value of

\int \nolimits_{ - 1}^1 \,\,\left[ {\,x\,} \right[1 + \sin \,\,\mathrm{\pi }x]\;dx

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

denotes the greatest integer)

View answer and explanation

Correct option: A

\mathrm{2}

Option A (2) is correct because the integral simplifies to 2 after considering the greatest integer function and the sine function within the given limits.

375 Medium

375 of 596

Analyzing +4 | -1

\int \nolimits_0^1 \,\,\log \,\,\left\{ {\,\,\sin \,\,\left( {\frac{{\pi x}}{2}} \right)\,\,} \right\}\,\,dx

is equal to

View answer and explanation

Correct option: B

- \,\,\log \,\,2

Option B (

\ - \,\,\log \,2

) is correct because the integral can be evaluated using a known result for the integral of log sin(x).

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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

376 Medium

376 of 596

Application +4 | -1

\int\limits_{0}^\pi x f\left(\cos ^2 x+\tan ^4 x\right) d x=k \int\limits_{0}^{\pi / 2} f\left(\cos ^2 x+\tan ^4 x\right) d x

, then the value of

k

View answer and explanation

Correct option: B

\pi

Option B (

\pi

) is correct because, after applying the definite integral property and simplification, the value of k is found to be

\pi

377 Medium

377 of 596

Application +4 | -1

If area bounded by the curves

x=a y^2

and

y=a x^2

is 1, then

a

equals-

View answer and explanation

Correct option: A

1 / \sqrt{3}

Option A (

1 / \sqrt{3}

) is correct because it is the value of 'a' that makes the area bounded by the given curves equal to 1.

378 Medium

378 of 596

Analyzing +4 | -1

The value of the integral

\int\limits_{0}^1 x \cot ^{-1}\left(1-x^2+x^4\right) d x

is:

View answer and explanation

Correct option: B

\frac{\pi}{4}-\log _e 2

Option D (

\frac{\pi}{4}-\frac{1}{2} \log _e 2

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

379 Medium

379 of 596

Application +4 | -1

The value of integral

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

View answer and explanation

Correct option: B

\mathrm{2}

Option B (2) is correct because since x + 2 is positive over the interval [-1, 1], the absolute value simplifies, and the integral becomes

\int_{-1}^{1} 1 dx = 2

380 Medium

380 of 596

Application +4 | -1

The area enclosed between the parabolas

y^2=4 x

and

x^2=4 y

View answer and explanation

Correct option: D

\frac{16}{3}

sq. unit

Option D (

\frac{16}{3}

sq. unit) is correct because the area enclosed between the parabolas

y^2=4x

and

x^2=4y

is found by integrating the difference of the functions from 0 to 4, which results in

\frac{16}{3}

381 Medium

381 of 596

Application +4 | -1

The value of

\int \limits_a^{a + \left( {\pi /2} \right)} \,\,\left( {{{\sin }^4}\,\,x\,\, + \,\,{{\cos }^4}\,\,x} \right)\,\,dx

, is

View answer and explanation

Correct option: A

\frac{{3\,\pi }}{8}

Option A is correct because the integral of a periodic function over one period is independent of the starting point, and the integral is evaluated to be 3π/8.

382 Medium

382 of 596

Application +4 | -1

Suppose that

F(x)

is an antiderivative of

f(x)-\frac{\sin x}{x}

, where

x>0

, then

\int_1^3 \frac{\sin 2 x}{x} d x

can be expressed as -

View answer and explanation

Correct option: A

F(6)-F(2)

Option A (

F(6)-F(2)

) is correct because using u-substitution and the given antiderivative definition, the integral transforms to F(6) - F(2).

383 Medium

383 of 596

Analyzing +4 | -1

Let

f

be a positive function. Let

I_1=\int_{1-k}^k(x) f(x(1-x)) d x ; I_2=\int_{1-k}^k f(x(1-x)) d x,

where

2 k-1>0

. Then,

\frac{I_2}{I_1}

View answer and explanation

Correct option: D

Option D (2) is correct because by using King's property and adding the integrals, we find that I2 is twice I1.

384 Medium

384 of 596

Application +4 | -1

The value of

\int\limits_{0}^2 x^{\left[x^2+1\right]} \mathrm{d} x

, where

[x]

is the greatest integer less than or equal to

x

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the integral evaluates to a more complex expression than the provided options.

385 Medium

385 of 596

Application +4 | -1

The value of

\int\limits_{0}^\pi x \log (\sin x) \mathrm{d} x

is (given that

\int\limits_{0}^{\pi / 2} \log \sin x \mathrm{~d} x=-\frac{\pi}{2} \log 2)

View answer and explanation

Correct option: A

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

Option B (6) is correct because by setting up equations based on the given information about pairs and triplets, we can solve for n.

386 Medium

386 of 596

Application +4 | -1

\lim\limits_{n \to \infty } \,\,\frac{{1\,\, + \,\,{2^4} + \,\,{3^4} + \,\,... + \,\,{n^4}}}{{{n^5}}}\,\, - \,\,\lim\limits_{n \to \infty } \,\,\frac{{1\,\, + \,\,{2^3}\,\, + \,\,{3^3}\,\, + \,\,... + \,\,{n^3}}}{{{n^5}}}

View answer and explanation

Correct option: D

\frac{1}{5}

Option D (

\frac{1}{5}

) is correct because it represents the difference between the integrals of x⁴ and 0, evaluated from 0 to 1.

387 Medium

387 of 596

Application +4 | -1

\int\limits_{0}^{10} e^{x-[x]} d x([\cdot]

denotes integral part

)

is equal to

View answer and explanation

Correct option: A

10(e-1)

Option A (10(e-1)) is correct because the integral simplifies to 10 times the integral from 0 to 1 of e^x, which evaluates to 10(e-1).

388 Medium

388 of 596

Analyzing +4 | -1

The value of

\int_0^4 \frac{\left(y^2-4 y+5\right) \sin (y-2) d y}{\left(2 y^2-8 y+1\right)}

View answer and explanation

Correct option: A

Option A (0) is correct because the integrand is an odd function after a suitable substitution, and the integral is evaluated over a symmetric interval.

389 Medium

389 of 596

Application +4 | -1

\int \nolimits_0^\pi \,\,\frac{1}{{1 + \sin \,\,x}}\,\,dx

is equal to

View answer and explanation

Correct option: B

The correct answer is 2 because it is the result of evaluating the definite integral using trigonometric substitution and limits.

390 Medium

390 of 596

Application +4 | -1

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

f(3)=\frac{1}{2}[\ln 5-\ln 6]

, then

f(4)

is:

View answer and explanation

Correct option: A

\frac{1}{2}[\ln 17-\ln 19]

Option A (

\frac{{1}}{{2}}[\ln 17 - \ln 19]

) is correct because it is the result of evaluating the definite integral with the given condition.

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