$\mathop \smallint \nolimits_1^x \,\,\frac{{\log \,\,\left( {{x^2}} \right)}}{x}\;\,\,dx$ is equal to

$\frac{1}{2}\,\,{(\log \,\,x)^2}$

The value of the integral $\mathop \smallint \limits_0^\pi \,\,\frac{{x\,dx}}{{1 + \cos \,\,\alpha \,\,\sin \,\,x}},\;\,0 < \alpha < \pi $, is

$\frac{{\pi \alpha }}{{1 + \sin \,\,\alpha }}$

$\frac{{\pi \alpha }}{{\cos \,\,\alpha }}$

$\frac{{\pi \alpha }}{{1 + \cos \,\,\alpha }}$

If $f\,\,\left( x \right) = \mathop \smallint \nolimits_{{x^2}}^{{x^2} + 1} \,\,{e^{ - {t^2}}}\,dt$, then $f\,\,\left( x \right)$ increases in

$\left( {0,\;\,\,\infty } \right)$

If $(\,\mathop \smallint \nolimits_0^a \,\,x\,dx\,)\;\,\, \le \,\,\,\left( {a + 4} \right)$, then

$a\,\,\, \le \,\, - 2$ or $a\,\,\, \ge \,\,4$

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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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511 Medium

511 of 596

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.

512 Medium

512 of 596

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.

513 Medium

513 of 596

Application +4 | -1

The value of

\int \limits_{ - \pi }^\pi \sin \,\,x\;\,\,f\left( {\cos \,\,x} \right)dx

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the integrand is an odd function, and the integral of an odd function over a symmetric interval is zero.

514 Medium

514 of 596

Analyzing +4 | -1

Let

f\;

be apositive function. Let

{I_1} = \int \nolimits_{1 - k}^k \,\,xf\left\{ {x\left( {1 - x} \right)} \right\},\,\,{I_2} = \int \nolimits_{1 - k}^k \,\,f\left\{ {x\left( {1 - x} \right)} \right\}

dx\;

where

2k\,\, - \,\,1 > 0

. Then

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

View answer and explanation

Correct option: C

\frac{1}{2}

Option C (1/2) is correct because by using the property ∫abf(x)dx = ∫abf(a+b-x)dx, we can show that I1 = 1/2 * I2, thus I1/I2 = 1/2.

515 Medium

515 of 596

Application +4 | -1

\int \nolimits_0^1 \,\,\frac{{x\,\,\;dx}}{{\left[ {x + \sqrt {1 - {x^2}} } \right]\,\,\sqrt {1 - {x^2}} }}

is equal to

View answer and explanation

Correct option: C

\frac{\pi }{4}

Option C (

\frac{\pi }{4}

) is correct because after trigonometric substitution and using properties of definite integrals, the integral simplifies to

\frac{\pi}{4}

516 Medium

516 of 596

Analyzing +4 | -1

f\left( t \right) = \int \nolimits_{ - t}^t \,\,\frac{{{e^{ - \left| x \right|}}}}{2}\,\,dx,\;\,\,\mathrm{then}\,\,\lim\limits_{t \to \infty } \,\,f\,\,\left( t \right)

is equal to

View answer and explanation

Correct option: A

\mathrm{1}

Option A (1) is correct because the limit of the integral as t approaches infinity is 1 after evaluating the integral and applying the limit.

517 Medium

517 of 596

Application +4 | -1

For

0 \leq x \leq \pi

, the area bounded by

y=x

and

y=x+\sin x

, is

View answer and explanation

Correct option: A

Option A (2) is correct because the area bounded by the two curves is calculated by integrating the difference of the functions from 0 to

\pi

, which simplifies to the integral of

\sin x

from 0 to

\pi

518 Medium

518 of 596

Analyzing +4 | -1

The value of the integral

\int \limits_{ - \pi /4}^{\pi /4} \,\,{\sin ^{ - 4}}\,x\,dx

, is

View answer and explanation

Correct option: A

- \frac{8}{3}

The correct answer is A (

- \frac{8}{3}

) because the integral of the given function over the specified limits is calculated to be -8/3.

519 Medium

519 of 596

Application +4 | -1

The area bounded by the curves

y = f\left( x \right)

, the

x

-axis and the ordinates

x = 1

and

x = b

\left( {b - 1} \right)\sin \left( {3b + 4} \right)

. Then,

f\left( x \right)

View answer and explanation

Correct option: C

\sin \left( {3x + 4} \right) + 3\left( {x - 1} \right)\cos \left( {3x + 4} \right)

Option C (sin(3x + 4) + 3(x - 1)cos(3x + 4)) is correct because it is the derivative of the given area function, representing f(x).

520 Medium

520 of 596

Application +4 | -1

The area in the first quadrant between

x^2+y^2=\pi^2

and

y=\sin x

View answer and explanation

Correct option: A

\frac{\left(\pi^3-8\right)}{4}

Option A (

\frac{\left(\pi^3-8\right)}{4}

) is correct because the area in the first quadrant between

x^2+y^2=\pi^2

and

y=\sin x

is the area of the quarter circle minus the area under the sine curve from 0 to

\pi

, which results in

\frac{\pi^3-8}{4}

521 Medium

521 of 596

Application +4 | -1

The value of the integral

\int_{-\pi}^\pi \sin m x \sin n x d x

for

m \neq n(m, n \in I)

View answer and explanation

Correct option: A

Option A (0) is correct because the integral of

\sin mx \sin nx

over

[-\pi, \pi]

is zero when

m \neq n

due to orthogonality.

522 Medium

522 of 596

Analyzing +4 | -1

The integral

\int \sqrt{1+2 \cot x(\operatorname{cosec} x+\cot x)} d x

\left(0 < x < \frac{\pi}{2}\right)

is equal to:
(where

\mathrm{C}

is a constant of integration)

View answer and explanation

Correct option: A

2 \log \left|\sin \frac{x}{2}\right|+C

Option A (

2 \log \left|\sin \frac{{x}}{{2}}\right|+C

) is correct because simplifying the integrand leads to an expression that can be directly integrated to obtain the given result.

523 Medium

523 of 596

Analyzing +4 | -1

The value of the definite integral

\int\limits_{\frac{-3 \pi}{4}}^{\frac{5 \pi}{4}} \frac{\cos x+\sin x}{1+e^{x-\frac{\pi}{4}}} d x

equals

View answer and explanation

Correct option: A

Option A (0) is correct because the integral evaluates to zero using properties of definite integrals.

524 Medium

524 of 596

Analyzing +4 | -1

Let

f(x)

be a function satisfying

f(x)+f(\pi-x)=\pi^{2}

\forall \, x \in \mathbb{R}

. Then

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

is equal to

View answer and explanation

Correct option: C

\pi^{2}

The correct answer is C (

\pi^{2}

) because using the property of definite integrals and the given functional equation, the integral simplifies to

\pi^{2}

after substitution and simplification.

525 Medium

525 of 596

Application +4 | -1

f(x)=a e^{2 x}+b e^x+c x, f(0)=-1, f(\ln 2)=4

, if

\int_0^{\ln 4}(f(x)-c x) d x=\frac{39}{2}

find

|a+b+c|

View answer and explanation

Correct option: A

Option A (25) is correct because by substituting the given conditions into the function and integral, we can solve for a, b, and c, and then find |a+b+c|.

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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

526 Medium

526 of 596

Application +4 | -1

a

is fixed real number such that

f\left( {a - x} \right)\,\, + \,\,f\left( {a + x} \right) = 0

, then

\int \limits_0^{2a} \,\,f\left( x \right)\,\,dx =

View answer and explanation

Correct option: B

\mathrm{0}

Option B (0) is correct because the integral of the function over the interval [0, 2a] is zero due to the property

f(a-x) + f(a+x) = 0

, which implies

f(2a-x) = -f(x)

527 Medium

527 of 596

Application +4 | -1

If [t] denotes the greatest integer

\leq t

, then the value of

\frac{3(e-1)}{e} \int\limits_{1}^2 x^2 e^{[x]+\left[x^3\right]} d x

View answer and explanation

Correct option: D

e^8-e

Option D (

e^8-e

) is correct because it is the result of evaluating the definite integral after applying the greatest integer function and simplifying.

528 Medium

528 of 596

Application +4 | -1

\int_{.}^b x^3 d x=0

and

\int_a^b x^2 d x=\frac{2}{3}

, then the value of

a

and

b

will be respectively

View answer and explanation

Correct option: D

-1,1

Option D (-1,1) is correct because it satisfies both integral equations, where the first integral being zero implies symmetry about the y-axis, and the second integral gives the specific values.

529 Medium

529 of 596

Application +4 | -1

The value of

\int \limits_0^\pi \,\,\frac{1}{{5\,\, + \,\,3\,\,\cos \,\,x}}\,\,dx

, is

View answer and explanation

Correct option: C

\pi \,\,/\,\,4

Option C (

\pi \\/\\4

) is correct because the definite integral is evaluated using trigonometric substitution and properties of definite integrals.

530 Medium

530 of 596

Application +4 | -1

\int \nolimits_1^x \,\,\frac{{\log \,\,\left( {{x^2}} \right)}}{x}\;\,\,dx

is equal to

View answer and explanation

Correct option: A

{(\log \,\,x)^2}

Option A (

{(\log \,\,x)^2}

) is correct because the integral of the given expression simplifies to this result after substitution.

531 Medium

531 of 596

Application +4 | -1

\int \frac{d x}{\cos x+\sqrt{3} \sin x}

equals

View answer and explanation

Correct option: C

\frac{1}{2} \log \tan \left(\frac{x}{2}+\frac{\pi}{12}\right)+C

Option C (

\frac{1}{2} \log \tan (\frac{x}{2}+\frac{\pi}{12})+C

) is correct because the integral can be transformed into a cosecant integral and then solved using the tangent half-angle formula.

532 Medium

532 of 596

Analyzing +4 | -1

I_n=\int\limits_{0}^{\pi / 4} \tan ^n x d x

then

\lim \limits_{{n \rightarrow \infty}} n\left(I_n+I_{n+2}\right)

equals

View answer and explanation

Correct option: B

The correct answer is B (1) because by using integration by parts and limit properties, the expression simplifies to 1.

533 Medium

533 of 596

Application +4 | -1

\int \nolimits_{\pi /6}^{\pi /3} \,\,\frac{{dx}}{{1 + \sqrt {\tan \,\,x} }}

is equal to

View answer and explanation

Correct option: A

\frac{\pi }{{12}}

Option A (

\frac{\pi }{{12}}

) is correct because using the property of definite integrals

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

simplifies the integral to

\frac{\pi}{12}

534 Medium

534 of 596

Application +4 | -1

The integral

\int\limits_ {\sqrt{\ln 5}}^{\sqrt{\ln 7}} \frac{x \cos x^2 d x}{\cos \left(\ell n 35-x^2\right)+\cos x^2}

is equal to

View answer and explanation

Correct option: A

\frac{1}{4} \ln \frac{7}{5}

Option B (

\frac{{1}}{{4}} \ln \frac{{7}}{{5}}

) is correct because it represents the value of the definite integral.

535 Medium

535 of 596

Application +4 | -1

The value of the integral

\int \limits_0^\pi \,\,\frac{{x\,dx}}{{1 + \cos \,\,\alpha \,\,\sin \,\,x}},\;\,0 < \alpha < \pi

, is

View answer and explanation

Correct option: A

\frac{{\pi \alpha }}{{\sin \,\alpha }}

Option A (

\frac{{\pi \alpha }}{{\sin \alpha }}

) is correct because the integral is solved using properties of definite integrals and trigonometric substitutions.

536 Medium

536 of 596

Application +4 | -1

A

is the area bounded by

x+2 y^2=0

and

x+3 y^2=1

, then find the value of

3 A

View answer and explanation

Correct option: A

Option A (4) is correct because the area bounded by the curves is calculated through integration, and multiplying the result by 3 gives 4.

537 Medium

537 of 596

Application +4 | -1

f(x)=\int\limits_{0}^x \cos ^4 t d t

, then

f(x+\pi)

equals

View answer and explanation

Correct option: A

f(x)+f(\pi)

Option A is correct because by using the property of definite integrals and trigonometric identities, we can show that f(x+

\pi

) = f(x) + f(

\pi

538 Medium

538 of 596

Analyzing +4 | -1

f\,\,\left( x \right) = \int \nolimits_{{x^2}}^{{x^2} + 1} \,\,{e^{ - {t^2}}}\,dt

, then

f\,\,\left( x \right)

increases in

View answer and explanation

Correct option: D

\left( { - \,\infty ,\;\,0} \right)

Option D ((

- \,\infty ,\;\,0

)) is correct because the derivative of the function is positive when x < 0, indicating that the function is increasing in that interval.

539 Medium

539 of 596

Application +4 | -1

(\,\int \nolimits_0^a \,\,x\,dx\,)\;\,\, \le \,\,\,\left( {a + 4} \right)

, then

View answer and explanation

Correct option: B

- \,2 \le \,\,\,a\,\, \le 4

Option B (

- \\,2 \le \\,\\,a\\,\le 4

) is correct because it represents the interval where the inequality holds true after solving the quadratic inequality.

540 Medium

540 of 596

Analyzing +4 | -1

(a, 0) ; a>0

is the point where the curve

y=\sin 2 x-\sqrt{3} \sin x

cuts the

X

-axis first,

A

is the area bounded by this part of the curve, the origin and the positive

X

-axis, then

View answer and explanation

Correct option: A

4 A+8 \cos a=7

Option A (

4 A+8 \cos a=7

) is correct because after finding the value of 'a' and the area A, substituting them into the equation yields a true statement.

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