The value of the integral $\mathop \smallint \limits_0^\pi \,\,\frac{1}{{{a^2} - \,\,2a\,\,\cos \,\,x\,\, + \,\,1}}\,\,dx\,\,(a < 1)$ is

$\frac{\pi }{{{a^2}\,\, - \,\,1}}$

$\frac{{2\;\pi }}{{{a^2}\,\, - \,\,1}}$

Let $f:(0,1) \in(0,1)$ be a differentiable function such that $f(x) \neq 0$ for all $x \in(0,1)$ and $f\left(\frac{1}{2}\right)=\frac{\sqrt{3}}{2}$. Suppose for all $x$, $\mathop {\lim }\limits_{t \to x} \,\,\,\frac{{\int_0^t {\sqrt {1 - {{(f(s))}^2}} } ds - \int_0^x {\sqrt {1 - {{(f(s))}^2}} } ds}}{{f(t) - f(x)}} = f(x)$ Then, the value of $f\left(\frac{1}{4}\right)$ belongs to

$\left\{\frac{\sqrt{7}}{2}, \frac{\sqrt{15}}{2}\right\}$

$\left\{\frac{\sqrt{7}}{3}, \frac{\sqrt{15}}{3}\right\}$

If $\smallint \,\,\frac{{{x^5}}}{{\sqrt {1 + {x^3}} }}\,\,dx$ is equal to

$\frac{2}{9}\,\,{\left( {1 + {x^3}} \right)^{5/2}}\,\, + \,\,\frac{2}{3}\,\,{\left( {1 + {x^3}} \right)^{3/2}}\,\, + \,\,C$

$\log \,\,\left| {\sqrt x + \sqrt {1 + {x^3}} } \right|\, + \,\,C$

${x^2}\,\,\log \,\,\left( {1 + {x^3}} \right)\,\, + \,\,C$

The value of $\mathop \smallint \nolimits_0^1 \,\,\frac{{{{\tan }^{ - 1}}\,\,x}}{{1 + \,\,{x^2}}}\;\,\,dx$ is

$\frac{{{{\rm{\pi }}^2}}}{{32}}$

$I = \left| {\,\mathop \smallint \nolimits_2^5 \,\,\frac{{\sin \,\,x\,\,dx}}{{(1 + {x^2}}}\,} \right|,\,\,{\rm{then}}$

$I$ lies in the interval $\left( {\frac{1}{4},\,\,\frac{1}{5}} \right)$

$I$ liesin the interval $\left( {\frac{1}{5},\,\,\frac{1}{6}} \right)$

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JEE • MATHEMATICS • DEFINITE INTEGRATION

Definite integration practice · Easy

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691 Easy

691 of 791

Application +4 | -1

The value of the integral

\int \limits_0^\pi \,\,\frac{1}{{{a^2} - \,\,2a\,\,\cos \,\,x\,\, + \,\,1}}\,\,dx\,\,(a < 1)

View answer and explanation

Correct option: A

\frac{\pi }{{1\,\, - \,\,{a^2}}}

Option A (

\frac{\pi }{{1\,\, - \,\,{a^2}}}

) is correct because after applying trigonometric identities and substitution, the integral evaluates to the given expression.

692 Easy

692 of 791

Application +4 | -1

\int_0^\pi x f(\sin x) d x

is equal to

View answer and explanation

Correct option: D

\pi \int_0^{\pi / 2} f(\cos x) d x

Option D is correct because the integral can be simplified using the property

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

and trigonometric identities to arrive at the given result.

693 Easy

693 of 791

Application +4 | -1

A

is the area between the curve

y = \sin x

and

x -

axis in the interval

\left[ {0,\pi /4} \right]

, then in the same interval, area between the curve

y = \cos x

and

x

-axis is

View answer and explanation

Correct option: C

1 - A

Option C (1 - A) is correct because the area under cos(x) from 0 to π/4 can be expressed as 1 - A, where A is the area under sin(x) in the same interval.

694 Easy

694 of 791

Application +4 | -1

f\,\,\left( x \right) = \tan x\,\, - \,\,{\tan ^3}x\,\, + \,\,{\tan ^5}\,\,x\,\, - ...\,\infty \,\,\mathrm{with}\,\,0\,\, < \,\,x\,\, < \,\,\frac{\pi }{4},\,\,\mathrm{then}\mathop {\,\,\int }\nolimits_0^{\pi /4} \,\,f\,\,\left( x \right)\,\,dx

is equal to

View answer and explanation

Correct option: C

\frac{1}{4}

Option C (

\frac{1}{4}

) is correct because the function simplifies to

\frac{{\sin \,2x}}{2}

, and its integral from 0 to

\frac{\pi}{4}

\frac{1}{4}

695 Easy

695 of 791

Analyzing +4 | -1

Let

{I_n}\left( x \right) = \int\limits_0^x {\frac{1}{{{{\left( {{t^2} + 5} \right)}^2}}}dt,n = 1,2,3.........\,.}

Then

View answer and explanation

Correct option: A

50 I_6-9 I_5=x I_5^{\prime}

Option A (

50 I_6-9 I_5=x I_5^{\prime}

) is correct because it results from applying integration by parts and simplifying the expression.

696 Easy

696 of 791

Application +4 | -1

The odd natural number

a

, such that the area of the region bounded by

y=1, y=3, x=0, x=y^a

\frac{364}{3}

, is equal to

View answer and explanation

Correct option: B

Option B (5) is correct because it satisfies the integral equation derived from the given area and limits.

697 Easy

697 of 791

Analyzing +4 | -1

Let

f:(0,1) \in(0,1)

be a differentiable function such that

f(x) \neq 0

for all

x \in(0,1)

and

f\left(\frac{1}{2}\right)=\frac{\sqrt{3}}{2}

. Suppose for all

x

\lim\limits_{t \to x} \,\,\,\frac{{\int_0^t {\sqrt {1 - {{(f(s))}^2}} } ds - \int_0^x {\sqrt {1 - {{(f(s))}^2}} } ds}}{{f(t) - f(x)}} = f(x)

Then, the value of

f\left(\frac{1}{4}\right)

belongs to

View answer and explanation

Correct option: D

\left\{\frac{\sqrt{7}}{4}, \frac{\sqrt{15}}{4}\right\}

The correct answer is

\left{\frac{\sqrt{7}}{4}, \frac{\sqrt{15}}{4}\right\}

because after solving the differential equation derived from the given limit and using the initial condition, we find

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

, leading to

f(\frac{1}{4}) = \frac{\sqrt{7}}{4}

698 Easy

698 of 791

Application +4 | -1

The value of

\int \nolimits_1^2 \,\,{\left[ {\,f\,\,\left\{ {g\,\,\left( x \right)} \right\}} \right]^{ - 1}}f'\left\{ {g\,\,\left( x \right)} \right\}\,\,g'\,\,\left( x \right)\,\,dx

, where

g\left( 1 \right) = g\left( 2 \right)

is equal to

View answer and explanation

Correct option: C

\mathrm{0}

Option C (

\mathrm{0}

) is correct because the integral evaluates to zero since the limits result in the same value due to g(1) = g(2).

699 Easy

699 of 791

Analyzing +4 | -1

\lim \limits_{{x \rightarrow \infty}} \frac{\int\limits_{0}^{2 x} t e^{t^2} d t}{e^{4 x^2}}

equals

View answer and explanation

Correct option: C

\frac{1}{2}

Option C (

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

) is correct because applying L'Hopital's rule simplifies the limit to

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

700 Easy

700 of 791

Application +4 | -1

The value of

\int \nolimits_{{e^{ - 1}}}^{{e^2}} \,\,\left| {\,\frac{{{{\log }_e}\,\,x}}{x}\,} \right|\,\,dx

View answer and explanation

Correct option: B

\frac{5}{2}

Option B (

\frac{5}{2}

) is correct because the integral is split into two parts based on the absolute value, and then solved using substitution.

701 Easy

701 of 791

Application +4 | -1

\int \nolimits_0^{\pi /8} \,\,{\cos ^3}\,\,4\theta \,\,d\theta

is equal to

View answer and explanation

Correct option: D

\frac{1}{6}

Option D (

\frac{1}{6}

) is correct because the integral of cos³(4θ) from 0 to π/8 evaluates to 1/6.

702 Easy

702 of 791

Application +4 | -1

\int\limits_{0}^{10}[x]^3\{x\} d x

is equal to:

View answer and explanation

Correct option: B

\frac{2025}{2}

Option B (

\frac{2025}{2}

) is correct because the integral is evaluated by splitting it into intervals and using the properties of greatest integer function and fractional part function.

703 Easy

703 of 791

Application +4 | -1

\lim \limits_{{n \rightarrow \infty}} \sum_{k=1}^n \frac{1}{\sqrt{n^2+3 k n}}

equal to

View answer and explanation

Correct option: B

\frac{2}{3}

Option B (

\frac{{2}}{{3}}

) is correct because it correctly converts the limit of the Riemann sum to a definite integral and evaluates it.

704 Easy

704 of 791

Application +4 | -1

\int\limits_{0}^1|3 x-1| d x

equals

View answer and explanation

Correct option: A

05-Jun

Option A (0.833) is correct because it is the result of evaluating the definite integral by splitting it at the point where the absolute value changes sign.

705 Easy

705 of 791

Application +4 | -1

The area bounded by the curve

y = {\sec ^2}x,\;y = 0

and

\left| x \right| = \frac{\pi }{3}

View answer and explanation

Correct option: C

2\sqrt 3

sq unit

Option C (

2\sqrt 3

sq unit) is correct because it represents the definite integral of

\sec^2 x

from

-\frac{\pi}{3}

\frac{\pi}{3}

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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

706 Easy

706 of 791

Analyzing +4 | -1

\lim \limits_\limits {n \rightarrow \infty}\left(\frac{(n+1)(n+2) \ldots 3 n}{n^{2 n}}\right)^{\frac{1}{n}}

is equal to:

View answer and explanation

Correct option: D

\frac{27}{\mathrm{e}^2}

Option D (

\frac{27}{\mathrm{e}^2}

) is correct because by converting the limit to an integral and evaluating, the result is

\frac{27}{\mathrm{e}^2}

707 Easy

707 of 791

Application +4 | -1

\int \limits_{\log 2}^x \,\,\frac{1}{{\sqrt {{e^x} - \,\,1} }}\,\,dx = \frac{\pi }{6}

is equal to

View answer and explanation

Correct option: C

\log \,\,4

Option C (

\log \,4

) is correct because after solving the definite integral with the given limits and substitution, we find that x =

\log \,4

708 Easy

708 of 791

Analyzing +4 | -1

\int \nolimits_{ - 1/2}^{1/2} \,\,\cos \,\,x\,\,\log \,\,\left( {\frac{{1 + x}}{{1 - x}}} \right)\,\,dx = k\,\,.\,\,\log \,\,2

, then

k

equals

View answer and explanation

Correct option: A

\mathrm{0}

Option A (

\mathrm{0}

) is correct because the integrand is an odd function, and the integral is evaluated over a symmetric interval.

709 Easy

709 of 791

Application +4 | -1

The value of the integral

\int \nolimits_0^{\pi /2} \,\,{\sin ^5}\,\,x\,dx

View answer and explanation

Correct option: C

\frac{8}{{15}}

Option C (

\frac{8}{{15}}

) is correct because it is the result of applying Walli's formula for the definite integral of

\sin^n x

from 0 to

\pi/2

when n is odd.

710 Easy

710 of 791

Application +4 | -1

\int \,\,\frac{{{x^5}}}{{\sqrt {1 + {x^3}} }}\,\,dx

is equal to

View answer and explanation

Correct option: B

\frac{2}{9}\,\,{\left( {1 + {x^3}} \right)^{3/2}}\,\, - \,\,\frac{2}{3}\,\,{\left( {1 + {x^3}} \right)^{1/2}} + \,\,C

Option B (

\frac{2}{9}\\ {\left( {1 + {x^3}} \right)^{3/2}}\\ - \\ \frac{2}{3}\\ {\left( {1 + {x^3}} \right)^{1/2}} + \\ C

) is correct because it is the result of the integration after applying the correct substitution and simplification.

711 Easy

711 of 791

Application +4 | -1

The value of

\int \nolimits_0^1 \,\,\frac{{{{\tan }^{ - 1}}\,\,x}}{{1 + \,\,{x^2}}}\;\,\,dx

View answer and explanation

Correct option: B

\frac{{{\mathrm{\pi }^2}}}{{32}}

Option B (

\frac{{{\mathrm{\pi }^2}}}{{32}}

) is correct because it is the result of a u-substitution and evaluating the resulting integral.

712 Easy

712 of 791

Analyzing +4 | -1

The function

F\left( x \right) = \int \limits_0^x \log \,\,\left( {\frac{{1 - x}}{{1 + x}}} \right)dx

, is

View answer and explanation

Correct option: A

An even function

Option A (An even function) is correct because the integral of an odd function from 0 to x results in an even function.

713 Easy

713 of 791

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: A

\frac{2}{\sqrt{5}} \tan ^{-1} \frac{1}{\sqrt{5}}

Option A (

\frac{2}{\sqrt{5}} \tan ^{-1} \frac{1}{\sqrt{5}}

) is correct because using the Weierstrass substitution simplifies the integral to a standard arctangent form.

714 Easy

714 of 791

Application +4 | -1

\{\mathrm{F}(\mathrm{x})\}^{101}=\int_0^{\mathrm{x}}\left(\mathrm{F}(\mathrm{t})^{100} \frac{\mathrm{dt}}{1+\sin \mathrm{t}}\right.

, then find

\mathrm{F}(\mathrm{x})

View answer and explanation

Correct option: B

\frac{\tan x-\sec x+1}{101}

Option B (

\frac{{\tan x - \sec x + 1}}{{101}}

) is correct because after differentiating and solving the differential equation with the initial condition, we obtain the given expression.

715 Easy

715 of 791

Application +4 | -1

The value of the integral

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

is equal to

View answer and explanation

Correct option: B

2 \pi

Option B (

2 \pi

) is correct because the expression inside the integral simplifies to

\frac{\pi}{2}

, and the integral of a constant is straightforward.

716 Easy

716 of 791

Application +4 | -1

The value of

\int \nolimits_0^{\pi /2} \,\,\frac{{{2^{\mathrm{sin}x}}}}{{{2^{\mathrm{sin}x}} + \,\,{2^{\mathrm{cos}x}}}}\;\,\,dx

View answer and explanation

Correct option: C

\mathrm{\pi }/4

Option C (

\mathrm{\pi }/4

) is correct because it is obtained by using the property of definite integrals and adding two forms of the integral to simplify and solve.

717 Easy

717 of 791

Analyzing +4 | -1

I = \left| {\,\int \nolimits_2^5 \,\,\frac{{\sin \,\,x\,\,dx}}{{(1 + {x^2}}}\,} \right|,\,\,\mathrm{then}

View answer and explanation

Correct option: D

I \le \frac{3}{{10}}

Option D (

I \le \frac{3}{{10}}

) is correct because the integral is bounded above by

\frac{3}{10}

by using properties of inequalities and integrals.

718 Easy

718 of 791

Application +4 | -1

The total area enclosed by the curve

a^2 y^2=x^2\left(a^2-x^2\right)

View answer and explanation

Correct option: A

\frac{4 a^2}{3}

Option A (

\frac{4 a^2}{3}

) is correct because it is the result of integrating the given function over the appropriate interval and multiplying by 4 due to symmetry.

719 Easy

719 of 791

Application +4 | -1

The value of

\int \nolimits_0^a \,\,\sqrt {\frac{{a - x}}{x}} \,\,dx

View answer and explanation

Correct option: C

\frac{{\pi a}}{2}

Option C (

\frac{{\pi a}}{2}

) is correct. The integral is solved using trigonometric substitution, leading to the value

\frac{\pi a}{2}

720 Easy

720 of 791

Application +4 | -1

f(x)=[|x|]

, then

\int\limits_{0}^{50} f(x) d x

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

View answer and explanation

Correct option: A

1225

Option A (1225) is correct because the integral is the sum of the integers from 0 to 49, which equals 1225.

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