The value of the $\smallint \,\,\frac{{\sin \,\,x\,\, + \,\,\cos \,\,x}}{{3\,\, + \,\,\sin \,\,2x}}\,\,dx$ is

$\frac{1}{2}\,\,\log \,\,\left( {\frac{{2\,\, + \,\,\sin \,\,x}}{{2\,\, - \,\,\sin \,\,x}}} \right) + c$

$\frac{1}{4}\,\,\log \,\,\left( {\frac{{1\,\, + \,\,\sin \,\,x}}{{1\,\, - \,\,\sin \,\,x}}} \right) + c$

$\mathop \smallint \nolimits_{ - \pi /3}^{\pi /3} \,\,\frac{{x\,\,\sin \,\,x}}{{{{\cos }^2}\,\,x}}\,\,dx$ is equal to

$\frac{1}{3}\,\,\left( {4\pi + 1} \right)$

$\frac{{4\pi }}{3}\,\, + \,\,\log \,\,\tan \,\,\frac{{5\pi }}{{12}}$

If $f: D \in R$ be such that $f(x) = \sqrt {\sin (\cos x)} + \ln \left( { - 2{{\cos }^2}x + 3\cos x - 1} \right){\rm{, }}\,\,{\rm{then }}$ $\int_{{x_1}}^{{x_2}} {\left[ {\cos x - \frac{1}{2}} \right]} \,\,dx$ is equal to, where $x_1, x_2 \equiv D$ and [.] denotes the greatest integer function,

$\frac{1}{2}\left(x_2-x_1\right)$

$\frac{1}{2}\left(x_1-x_0\right)$

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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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421 Medium

421 of 596

Application +4 | -1

The value of the

\int \,\,\frac{{\sin \,\,x\,\, + \,\,\cos \,\,x}}{{3\,\, + \,\,\sin \,\,2x}}\,\,dx

View answer and explanation

Correct option: A

\frac{1}{4}\,\,\log \,\,\left( {\frac{{2 - \sin \,\,x\,\, + \,\,\cos \,\,x}}{{2 + \sin \,\,x\,\, - \,\,\cos \,\,x}}} \right) + c

Option A (

\frac{1}{4}\log \left( {\frac{{2 - \sin \\,x\\, + \\,\cos \\,x}}{{2 + \sin \\,x\\, - \\,\cos \\,x}}} \right) + c

) is correct because it is the result of the integration after a suitable substitution.

422 Medium

422 of 596

Application +4 | -1

The volume of the solid obtained by revolving about

y

-axis the area enclosed between the ellipse

{x^2} + 9{y^2} = 9

and the straight line

x + 3y = 3,\;

in the first quadrant is

View answer and explanation

Correct option: A

3\mathrm{\pi }

Option A,

3\mathrm{\pi }

, is the correct volume because it is found by subtracting the volume generated by revolving the line from the volume generated by revolving the ellipse.

423 Medium

423 of 596

Analyzing +4 | -1

The value of

\lim _{n \rightarrow \infty} \sum\limits_{{r=1}}^{r=4 n} \frac{\sqrt{n}}{\sqrt{r}(3 \sqrt{r}+4 \sqrt{n})^2}

is equal to

View answer and explanation

Correct option: C

\frac{1}{10}

Option C (

\frac{1}{10}

) is correct because the limit of the Riemann sum is evaluated by converting it to a definite integral and then solving the integral.

424 Medium

424 of 596

Analyzing +4 | -1

The value of the integral

\int\limits_{{-\log _e 2}}^{\log _2 2} e^x\left(\log _e\left(e^x+\sqrt{1+e^{2 x}}\right)\right) d x

is equal to

View answer and explanation

Correct option: A

\log _e\left(\frac{\sqrt{2}(2+\sqrt{5})^2}{\sqrt{1+\sqrt{5}}}\right)-\frac{\sqrt{5}}{2}

Option A is correct. The integral is solved by substitution and integration by parts, resulting in the given logarithmic expression.

425 Medium

425 of 596

Application +4 | -1

Let

f(x)=\int_1^x \sqrt{2-t^2} d t

. Then the real roots of the equation

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

are

View answer and explanation

Correct option: A

\pm 1

Option A (

\pm 1

) is the correct answer because it satisfies the equation derived from the given integral and its derivative.

426 Medium

426 of 596

Application +4 | -1

\int \nolimits_0^{2\pi } \,\,(\sin \,\,x\,\, + \,\,|\sin \,\,x)\,\,dx

is equal to

View answer and explanation

Correct option: B

\mathrm{4}

Option B (4) is correct because the integral of |sin x| from 0 to 2π is 4, and the integral of sin x from 0 to 2π is 0.

427 Medium

427 of 596

Application +4 | -1

f\left(\frac{3 x-4}{3 x+4}\right)=x+2, x \neq-\frac{4}{3}

, and

\int f(x) d x=\mathrm{A} \log |1-x|+\mathrm{B} x+\mathrm{C}

, then the ordered pair

(A, B)

is equal to :
(where

C

is a constant of integration)

View answer and explanation

Correct option: B

\left(-\frac{8}{3}, \frac{2}{3}\right)

Option B

\left(-\frac{8}{3}, \frac{2}{3}\right)

is correct because after finding f(x) and integrating, the coefficients A and B match these values.

428 Medium

428 of 596

Analyzing +4 | -1

I_1=\int\limits_{0}^1\left(1-x^4\right)^7 d x

and

I_2=\int\limits_{0}^1\left(1-x^4\right)^6 d x

then

\frac{29}{4} \frac{I_1}{I_2}

is equal to:

View answer and explanation

Correct option: C

Option C (7) is correct because it is the simplified value obtained after applying integration by parts and solving for the ratio of the two integrals.

429 Medium

429 of 596

Application +4 | -1

Let

f

be a continuous function satisfying

\int\limits_{0}^{t^2}\left(f(x)+x^2\right) d x=\frac{4}{3} t^3, \forall t>0

. Then

f\left(\frac{\pi^2}{4}\right)

is equal to

View answer and explanation

Correct option: C

\pi\left(1-\frac{\pi^3}{16}\right)

Option C (

\pi(1-\frac{\pi^3}{16})

) is correct because it is the value of the function

f(x)

evaluated at

x = \frac{\pi^2}{4}

after solving the given integral equation.

430 Medium

430 of 596

Analyzing +4 | -1

f(x)=x+\int_0^1\left(x y^2+x^2 y\right) f(y) d y

and

x+\frac{\left(61 x+80 x^2\right)}{\lambda}

, the

\mathrm{n}

the value of

\lambda

must be:

View answer and explanation

Correct option: A

119

Option A (119) is correct because by solving the integral equation, we find the function f(x) and then compare it to the given form to determine the value of lambda.

431 Medium

431 of 596

Application +4 | -1

\int\left\{\frac{(\log x-1)}{1+(\log x)^2}\right\}^2 d x

is equal to

View answer and explanation

Correct option: D

\frac{x}{(\log x)^2+1}+C

Option D (

\frac{{x}}{{(\log x)^2+1}}+C

) is correct because the integral can be solved by recognizing the form of the integrand after substitution.

432 Medium

432 of 596

Analyzing +4 | -1

The value of the definite integral

\int_0^{\infty} \frac{d x}{\left(1+x^a\right)\left(1+x^2\right)}(a>0)

View answer and explanation

Correct option: A

\frac{\pi}{4}

Option A (

\frac{\pi}{4}

) is correct because using the substitution

x = \tan \theta

, the integral simplifies to

\frac{\pi}{4}

, independent of the value of

a

433 Medium

433 of 596

Analyzing +4 | -1

\phi(x)=\frac{1}{\sqrt{x}} \int\limits_{\frac{\pi}{4}}^{x}\left(4 \sqrt{2} \sin t-3 \phi^{\prime}(t) d t, x > 0\right.

, then

\phi^{\prime}\left(\frac{\pi}{4}\right)

is equal to

View answer and explanation

Correct option: A

\frac{8}{6+\sqrt{\pi}}

The correct answer is A (

\frac{8}{6+\sqrt{\pi}}

) because it results from differentiating the given equation and substituting x =

\frac{\pi}{4}

434 Medium

434 of 596

Application +4 | -1

h\left( a \right) = h\left( b \right)

, the value of the integral

\int \limits_a^b \,\,{\left[ {f\,\,\left( {g\;\left( {h\,\,\left( x \right)\,} \right)\,} \right)} \right]^{ - 1}}\,\,f'\,\,\left( {\,g\,\,\left( {h\,\,\left( x \right)\,} \right)\,} \right)\,\,g'\,\,\left( {h\,\left( x \right)\,} \right)\,\,h'\,\,\left( x \right)\,\,dx

is equal to

View answer and explanation

Correct option: A

\mathrm{0}

Option A (

\mathrm{0}

) is correct because the limits of integration become equal, making the integral zero.

435 Medium

435 of 596

Application +4 | -1

The value of definite integral

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

is:

View answer and explanation

Correct option: C

{\pi ^2}

Option D (

{\pi ^2}

) is correct because it is the value of the definite integral after applying properties of definite integrals and trigonometric substitutions.

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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

436 Medium

436 of 596

Application +4 | -1

\phi '\,\,\left( x \right) = \frac{{{{\log }_e}\,\,\left| {\,\sin \,\,x\,} \right|}}{x},\;\,x \ne n\;\pi ,\,\;n \in Z

and

\int \limits_1^3 \,\,\frac{{3\,\,{{\log }_e}\,\left| {\,\sin \,\,{x^3}\,} \right|}}{x}\,\,dx = \phi \,\,\left( 1 \right)

, then the possible value of

k

View answer and explanation

Correct option: A

\mathrm{27}

Option A (27) is correct because the integral is evaluated using substitution and the properties of definite integrals, resulting in k = 27.

437 Medium

437 of 596

Analyzing +4 | -1

For any

n \in N

, the value of the integral

\int_0^\pi \frac{\sin 2 n x}{\sin x} d x

, is

View answer and explanation

Correct option: D

none of these

Option D (none of these) is correct because the value of the integral is 0 for all n in N, which is not among the provided options.

438 Medium

438 of 596

Analyzing +4 | -1

\left| {\mathop {\,\int }\limits_{10}^{19} \,\,\frac{{\sin \,\,x}}{{1 + {x^8}}}\,\,dx\,} \right|

is less than

View answer and explanation

Correct option: C

{10^{ - 7}}

Option C is correct because the absolute value of the integral is less than

{10^{ - 7}}

after bounding the integrand.

439 Medium

439 of 596

Analyzing +4 | -1

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

is equal to

View answer and explanation

Correct option: B

\frac{{4\pi }}{3}\,\, - \,\,2\,\,\log \,\,\tan \,\,\frac{{5\pi }}{{12}}

Option B (

\frac{{4\pi }}{3}\, - \,2\,\log \,\tan \,\frac{{5\pi }}{{12}}

) is correct because after applying integration by parts and simplifying, the result matches this option.

440 Medium

440 of 596

Application +4 | -1

f:R \to R

is given by

f(x) = x + 1

, then the value of

\lim\limits_{n \to \infty } \frac{1}{n}\left[ {f(0) + f\left( {\frac{5}{n}} \right) + f\left( {\frac{{10}}{n}} \right) + ..... + f\left( {\frac{{5(n - 1)}}{n}} \right)} \right]

View answer and explanation

Correct option: D

\frac{7}{2}

Option D (7/2) is correct because the limit can be expressed as a definite integral, which evaluates to 7/2.

441 Medium

441 of 596

Application +4 | -1

The value of

\int \limits_{ - 1}^2 \,\,\frac{{\left| {\,x\,} \right|}}{x}\,\,dx

, is

View answer and explanation

Correct option: B

\mathrm{1}

Option B (1) is correct because the integral of |x|/x from -1 to 2 can be split into two integrals, one from -1 to 0 and another from 0 to 2, and then evaluated.

442 Medium

442 of 596

Application +4 | -1

f(x)=x-[x]

: for every real number of

x

, when

[x]

is the integral point of

x

. Then

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

is equal to

View answer and explanation

Correct option: A

Option A (1) is correct because the integral of x-[x] from -1 to 1 simplifies to 1 after evaluating the integral part by part.

443 Medium

443 of 596

Understanding +4 | -1

\int \frac{\log \left(t+\sqrt{1+t^2}\right)}{\sqrt{1+t^2}} d t=\frac{1}{2}(g(t))^2+C

, where

\mathrm{C}

is a constant, then

g(b)

is equal to :

View answer and explanation

Correct option: D

\log (2+\sqrt{5})

Option D (

\log (2+\sqrt{5})

) is correct. By substituting

u = \log(t + \sqrt{1+t^2})

, the integral simplifies, and g(t) can be found.

444 Medium

444 of 596

Analyzing +4 | -1

f: D \in R

be such that

f(x) = \sqrt {\sin (\cos x)} + \ln \left( { - 2{{\cos }^2}x + 3\cos x - 1} \right)\mathrm{, }\,\,\mathrm{then }

\int_{{x_1}}^{{x_2}} {\left[ {\cos x - \frac{1}{2}} \right]} \,\,dx

is equal to, where

x_1, x_2 \equiv D

and [.] denotes the greatest integer function,

View answer and explanation

Correct option: A

Option A (0) is correct because the greatest integer function evaluates to 0 within the domain, making the integral zero.

445 Medium

445 of 596

Application +4 | -1

Let

F(x)

be the area bounded by the curve

f(t)=\frac{e^t}{t}

between

t=a(a > 1), t=x

and axis of abscissa then the area bounded by

g(t)=\frac{e^t}{t+t_0}\left(t_0 > 0\right)

between

t=a

t=x

and axis of abscissa is

View answer and explanation

Correct option: C

e^{-t_0}\left[F\left(x+t_0\right)-F\left(a+t_0\right)\right]

Option C (

e^{-t_0}[F(x+t_0)-F(a+t_0)]

) is correct because it represents the area calculated using a substitution and the given function F(x).

446 Medium

446 of 596

Analyzing +4 | -1

\int\limits_{0}^\pi \frac{x^2}{(1+\sin x)^2} d x=A

then

\int\limits_{0}^\pi \frac{2 x^2 \cos ^2(x / 2)}{(1+\sin x)^2} d x=

View answer and explanation

Correct option: D

A+2 \pi-\pi^2

Option D (

A+2 \pi-\pi^2

) is correct because after applying integration by parts and properties of definite integrals, the given integral simplifies to this expression.

447 Medium

447 of 596

Understanding +4 | -1

\int\limits_{{-\pi / 2}}^{\pi / 2} \log \left(\frac{2-\sin \theta}{2+\sin \theta}\right) d \theta=

View answer and explanation

Correct option: A

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

448 Medium

448 of 596

Application +4 | -1

\int_0^1 \sqrt{1-\cos x}

is equal to

View answer and explanation

Correct option: C

2 \sqrt{2}

Option C (

2 \sqrt{2}

) is correct because the integral of

\sqrt{1 - \cos x}

from 0 to

\pi

simplifies to

\sqrt{2} \int_0^{\pi} \sin \frac{x}{2} dx

, which evaluates to

2\sqrt{2}

449 Medium

449 of 596

Analyzing +4 | -1

The region bounded by the curve

x=\frac{1}{2}, x=2, y=\log x

and

y=2^x

, then the area of the region, is

View answer and explanation

Correct option: B

\frac{(4+\sqrt{2})}{\log 2}-\frac{5}{2} \log 2+\frac{17}{2}

sq unit

Option B (

\frac{{(4+\sqrt{{2}})}}{\log 2}-\frac{{5}}{{2}} \log 2+\frac{{17}}{{2}}

sq unit) is correct because it represents the area calculated by integrating the difference between the two functions over the given interval.

450 Medium

450 of 596

Understanding +4 | -1

n \in \mathbf{N}

, the value of

\int\limits_{0}^n[x] \mathrm{d} x

(where

[x]

is the greatest integer function) is

View answer and explanation

Correct option: B

\frac{n(n-1)}{2}

Option B (

\frac{{n(n-1)}}{{2}}

) is correct because the integral of the greatest integer function from 0 to n can be expressed as a sum of integers from 0 to n-1.

Revision notes and key takeaways

Review DEFINITE INTEGRATION in your textbook or class notes, then use this set to check your understanding.

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