The value of the integral $\mathop \smallint \limits_1^3 \,\,\sqrt {3\,\, + \,\,{x^3}} \,\,dx$ lies in the interval

$\begin{array}{*{20}{l}} {\left( {{\rm{1,}}\,\,{\rm{3}}} \right)} \end{array}$

For any natural number $n$, the value of the integral $\mathop \smallint \limits_0^{\sqrt n } \,\,\left[ {{x^2}} \right]\,\,dx$, is

$n\,\sqrt n \,\, - \,\,\mathop \sum \limits_{r = 1}^n \,\sqrt r $

$n\,\sqrt n \,\, + \,\,\mathop \sum \limits_{r = 1}^n \,\sqrt r $

$\mathop \sum \limits_{r = 1}^n \,\,\sqrt r \,\, - \,\,n\,\sqrt n $

If $f\left( x \right) = A\,\,\sin \,\,\left( {\frac{{\pi x}}{2}} \right) + B,\;\,\,f'\,\,\left( {\frac{1}{2}} \right) = \sqrt 2 $ and $\mathop \smallint \limits_0^1 \,f\left( x \right)dx = \frac{{2\,A}}{\pi }\,$, then the constants $A$ and $B$ are respectively

$\frac{4}{\pi }$ and ${\rm{0}}$

$\frac{\pi }{2}$ and $\frac{\pi }{2}$

$\frac{2}{\pi }$ and $\frac{3}{\pi }$

${\rm{0}}$ and $ - \frac{4}{\pi }$

If $f\left( x \right) = \mathop {\lim }\limits_{n \to \infty } \,\,{n^2}({x^{1/n}}\,\, - \,\,{x^{1/\left( {n + 1} \right)}}$, $x > 0,\,\;{\rm{then}}\,\,\smallint x\;f\left( x \right)\,\,dx$ is equal to

${x^2}\,\,\log \,\,x\,\, - \,\,\frac{1}{2}\,\,{x^2} + \,\,c$

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JEE • MATHEMATICS • DEFINITE INTEGRATION

Definite integration practice · Medium

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Showing 541-570 of 596 questions

541 Medium

541 of 596

Application +4 | -1

Let

k(x)

be a continuous function satisfying the equation

\int\limits_{0}^{x^3} k(t) d t=x^{1+x^2}

, find the value of

3 k(1)

View answer and explanation

Correct option: A

Option A (2) is correct because it's derived from evaluating the limit by converting the sum to an integral and then performing integration by parts.

542 Medium

542 of 596

Application +4 | -1

Let

\mathrm{f}:(-1,1) \rightarrow \mathrm{R}

be a continuous function. If

\int\limits_{0}^{\sin x} f(t) d t=\frac{\sqrt{3}}{2} x

, then

f\left(\frac{\sqrt{3}}{2}\right)

is equal to :

View answer and explanation

Correct option: D

\sqrt{3}

Option D (

\sqrt{3}

) is correct because it is the value obtained by differentiating the given integral equation using the Leibniz rule and substituting the appropriate value of x.

543 Medium

543 of 596

Application +4 | -1

f(x)=\int\limits_{{1 / x}}^{\sqrt{x}} \cos t^2 d t(x>0)

, then

\frac{d f(x)}{d x}

View answer and explanation

Correct option: B

\frac{x \sqrt{x} \cos x+2 \cos \left(x^{-2}\right)}{2 x^2}

Option B is correct because the derivative is calculated using the Leibniz rule for differentiating integrals.

544 Medium

544 of 596

Analyzing +4 | -1

The value of the integral

\int \limits_1^3 \,\,\sqrt {3\,\, + \,\,{x^3}} \,\,dx

lies in the interval

View answer and explanation

Correct option: C

\left( {4,\,\,2\,\,\sqrt {300} } \right)

The correct answer is C (4, 2

\sqrt{30}

) because the integral is bounded by the minimum and maximum values of the integrand over the interval.

545 Medium

545 of 596

Application +4 | -1

The value of the integral

\int\limits_{{-\pi}}^\pi(\cos a x-\sin b x)^2 d x,(a

and

b

are integer) is

View answer and explanation

Correct option: D

2 \pi

Option D (

2 \pi

) is correct because the integral can be simplified using trigonometric identities and properties of definite integrals.

546 Medium

546 of 596

Application +4 | -1

\int\limits_{0}^{1000} e^{x-[x]} d x

View answer and explanation

Correct option: C

1000(e-1)

Option C (

1000(e-1)

) is correct because the integral can be simplified using the periodicity of the function.

547 Medium

547 of 596

Application +4 | -1

\int\limits_{\frac{3 \sqrt{2}}{4}}^{\frac{3 \sqrt{3}}{4}} \frac{48}{\sqrt{9-4 x^{2}}} d x

is equal to

View answer and explanation

Correct option: D

2 \pi

Option D (

2 \pi

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

548 Medium

548 of 596

Application +4 | -1

Let

f(x)

be a non-negative continous function such that the area bounded by the curve

y=f(x), x

-axis and the ordinates

x=\frac{\pi}{4}, x=\beta > \frac{\pi}{4}

\left(\beta \sin \beta+\frac{\pi}{4} \cos \beta+\sqrt{2} \beta\right)

. Then

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

View answer and explanation

Correct option: B

\left(1-\frac{\pi}{4}+\sqrt{2}\right)

Option B is correct because by differentiating the given integral equation and substituting

\beta = \frac{\pi}{2}

, we can find the value of

f(\frac{\pi}{2})

549 Medium

549 of 596

Analyzing +4 | -1

A

denotes the area bounded by

f(x)=\left|\frac{\sin x+\cos x}{x}\right|, X

-axis,

x=\pi

and

x=3 \pi

, then

View answer and explanation

Correct option: B

0 < A < 2

Option B (

0 < A < 2

) is correct because the area A is bounded by the given function and X-axis between the specified limits, and its value falls within this range.

550 Medium

550 of 596

Analyzing +4 | -1

Let

f

be integrable over

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

for any real

a

. If we define

{I_1} = \int \nolimits_0^{\pi /2} \,\,\cos \,\,\mathrm{\theta }\,\,f(\sin \,\,\mathrm{\theta }\,\, + \,\,{\cos ^2}\,\,\mathrm{\theta })\,\,\;d\mathrm{\theta }

and

{I_2} = \int \nolimits_0^{\pi /2} \sin \,\,2\mathrm{\theta }\,\,f(\sin \,\,\mathrm{\theta }\,\, + \,\,{\cos ^2}\,\,\mathrm{\theta })\;\,\,d\mathrm{\theta }

, then

View answer and explanation

Correct option: A

{I_1} = {I_2}

Option A is correct because by substituting sin(

\theta

) + cos²(

\theta

) = t, the integral I1 - I2 becomes zero, implying I1 = I2.

551 Medium

551 of 596

Application +4 | -1

The integral

\int\limits_{0}^{\frac{\pi}{4}} \frac{136 \sin x}{3 \sin x+5 \cos x} d x

is equal to:

View answer and explanation

Correct option: A

3 \pi-50 \log _e 2+20 \log _e 5

The correct answer is A (

3 \pi-50 \log _e 2+20 \log _e 5

) because it is the result of evaluating the definite integral after expressing the integrand in a suitable form.

552 Medium

552 of 596

Application +4 | -1

\int\limits_ {\frac{-\pi}{2}}^{\frac{\pi}{2}} \frac{|x| d x}{8 \cos ^2 2 x+1}

has the value

View answer and explanation

Correct option: B

\frac{\pi^2}{12}

Option B

\frac{{\pi^2}}{{12}}

is correct because the integral of the given even function is evaluated using properties of definite integrals and trigonometric substitutions.

553 Medium

553 of 596

Application +4 | -1

If the area above the x-axis bounded by the curves

y = {2^{kx}}\mathrm{\;and\;}x = 0\;\,\mathrm{and\;}\,2\;is\,\frac{3}{{\log 2\;}}\;\mathrm{then\;the\;value\;of\;}k\mathrm{\;is}

View answer and explanation

Correct option: B

Option B, 1, is the correct value of k because when k = 1, the area under the curve y = 2^(kx) from x = 0 to x = 2 equals 3/log(2).

554 Medium

554 of 596

Analyzing +4 | -1

The value of the integral

I=\int\limits_{0}^a \frac{x^4}{\left(a^2+x^2\right)^4} d x

, is

View answer and explanation

Correct option: A

\frac{1}{16 a^3}\left(\frac{\pi}{4}-\frac{1}{3}\right)

Option A (

\frac{1}{16 a^3}(\frac{\pi}{4}-\frac{1}{3})

) is correct because it is obtained by using trigonometric substitution and simplifying the integral.

555 Medium

555 of 596

Application +4 | -1

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

View answer and explanation

Correct option: C

\frac{\pi}{4}

Option C (

\frac{\pi}{4}

) is correct because using the property

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

simplifies the integral to a readily solvable form.

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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

556 Medium

556 of 596

Application +4 | -1

If f(x)=f(a+x) and

\int \nolimits_0^a \,\,f\left( x \right)\,\,dx = k,\,\,\mathrm{then}\,\,\int \nolimits_0^{na} \,\,f\left( x \right)\,\,dx

is equal to

View answer and explanation

Correct option: A

n\,k

Option A (n k) is correct because the integral from 0 to na of f(x) dx is n times the integral from 0 to a of f(x) dx, given that f(x) = f(a+x).

557 Medium

557 of 596

Application +4 | -1

f\left( x \right) = \left\{ {\begin{array}{cccccccccccccccccccc} {x,\,\;\;for\;\,\,x\,\, < 1}\\ {x - 1,\;\,\,for\;\,\,x\,\, \ge 1} \end{array}} \right.

, then

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

is equal to

View answer and explanation

Correct option: C

\frac{5}{3}

Option C (

\frac{5}{3}

) is correct because it is the result of correctly evaluating the definite integral by splitting it into two integrals based on the piecewise definition of f(x).

558 Medium

558 of 596

Analyzing +4 | -1

7. The value of

\frac{e^{-\frac{\pi}{4}}+\int\limits_{{0}}^{\frac{\pi}{4}} e^{-x} \tan ^{50} x d x}{\int\limits_{{0}}^{\frac{\pi}{4}} e^{-x}\left(\tan ^{49} x+\tan ^{51} x\right) d x}

View answer and explanation

Correct option: B

Option B, 50, is correct because the integral simplifies to 50 after applying integration by parts and recognizing the pattern.

559 Medium

559 of 596

Analyzing +4 | -1

The value of

\int\limits_{0}^{\pi / 2}(2 \log \sin x-\log \sin 2 x) d x

, is

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because it is found by using properties of logarithms and definite integrals to simplify and evaluate the integral.

560 Medium

560 of 596

Analyzing +4 | -1

The value of

\int\limits_{{-\pi / 4}}^{\pi / 4} \frac{(\pi-4 \theta) \tan \theta}{1-\tan \theta} \mathrm{d} \theta

is equal to :

View answer and explanation

Correct option: C

\pi \ln 2-\frac{\pi^2}{4}

Option C (\

\pi \ln 2 - \frac{{\pi^2}}{{4}}

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

561 Medium

561 of 596

Understanding +4 | -1

f(x)

is a continuous periodic function with period

T

, then the integral

I=\int\limits_{a}^{a+T} f(x) d x

View answer and explanation

Correct option: C

Independent of a

The correct answer is Independent of a because the integral of a periodic function over one period is constant regardless of the starting point.

562 Medium

562 of 596

Application +4 | -1

\lim \limits_{{n \rightarrow \infty}}\left[\sin \frac{n}{n^2+1^2}+\sin \frac{n}{n^2+1^2}+\ldots+\sin \frac{n}{n^2+1^2}\right]=

View answer and explanation

Correct option: A

\frac{\pi}{4}

Option A (

\frac{\pi}{4}

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

\frac{\pi}{4}

563 Medium

563 of 596

Application +4 | -1

For any natural number

n

, the value of the integral

\int \limits_0^{\sqrt n } \,\,\left[ {{x^2}} \right]\,\,dx

, is

View answer and explanation

Correct option: B

n\,\sqrt n \,\, - \,\,\sum \limits_{r = 1}^n \,\sqrt r

Option D (

n\sqrt n - \sum \limits_{r = 1}^n \,\sqrt r

) is correct because the integral is evaluated by splitting it into a sum of integrals and simplifying.

564 Medium

564 of 596

Application +4 | -1

f\left( x \right) = A\,\,\sin \,\,\left( {\frac{{\pi x}}{2}} \right) + B,\;\,\,f'\,\,\left( {\frac{1}{2}} \right) = \sqrt 2

and

\int \limits_0^1 \,f\left( x \right)dx = \frac{{2\,A}}{\pi }\,

, then the constants

A

and

B

are respectively

View answer and explanation

Correct option: D

\frac{4}{\pi }

and

\mathrm{0}

Option D (

\frac{4}{\pi }

and 0) is correct. By using the given derivative and integral conditions, we can solve for the constants A and B.

565 Medium

565 of 596

Analyzing +4 | -1

The integral

\int\limits_{0}^\pi \sqrt{1+4 \sin ^2 \frac{x}{2}-4 \sin \frac{x}{2}} d x

equals:

View answer and explanation

Correct option: B

4 \sqrt{3}-4-\frac{\pi}{3}

Option B (

4 \sqrt{3}-4-\frac{\pi}{3}

) is correct because it is the result of evaluating the integral after splitting it into two parts based on the absolute value.

566 Medium

566 of 596

Analyzing +4 | -1

f\left( x \right) = \lim\limits_{n \to \infty } \,\,{n^2}({x^{1/n}}\,\, - \,\,{x^{1/\left( {n + 1} \right)}}

x > 0,\,\;\mathrm{then}\,\,\int x\;f\left( x \right)\,\,dx

is equal to

View answer and explanation

Correct option: D

None of these

Option B (

\frac{\mathrm{\pi }}{4}

) is correct because it is the result of the definite integral after the trigonometric substitution and simplification.

567 Medium

567 of 596

Application +4 | -1

I=\int\limits_{0}^1 \frac{e^t}{t+1} d t

, then

\int\limits_{{a-1}}^a \frac{e^{-t}}{t-a-1} d t=

View answer and explanation

Correct option: C

(-I) e^{-a}

Option C ((-I)

e^{-a}

) is correct because after substitution and simplification, the integral transforms to

(-I) e^{-a}

568 Medium

568 of 596

Application +4 | -1

Let

f(x)=\int_2^x \frac{d t}{\sqrt{1+t^4}}

and

g

be the inverse of then the value of

g^{\prime}(0)

View answer and explanation

Correct option: C

\sqrt{17}

Option C (

\sqrt{17}

) is correct because it is found by using the inverse function derivative formula and evaluating at the appropriate point.

569 Medium

569 of 596

Application +4 | -1

The value of

\int\limits_{0}^\pi|\cos x| d x

, is

View answer and explanation

Correct option: B

Option B (2) is correct because the integral of the absolute value of cos(x) from 0 to pi is equal to 2.

570 Medium

570 of 596

Application +4 | -1

The area bounded by the curves

y^2=8 x

and

y=x

View answer and explanation

Correct option: B

\frac{32}{3} s q

. Unit

Option B (

\frac{32}{3} sq. Unit) is correct because the area bounded by the curves

y²=8x

and

y=x

is found by integrating the difference of the functions from 0 to 8, which results in

\frac{32}{3}

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