$\mathop \smallint \limits_1^{\frac{{4\sqrt 3 }}{5} - 1} \,\,\frac{{x + 2}}{{\sqrt {{x^2} + 2x - 3} }}\,\,dx$ equals to

$\frac{{2\,\sqrt 3 }}{3}\,\, - \,\,\frac{1}{2}\,\,\log \,\,\left( {\sqrt 3 + 2} \right)$

$\frac{{2\,\sqrt 3 }}{3}\,\, + \,\,\frac{1}{2}\,\,\log \,\,\left( {\sqrt 3 + 2} \right)$

$\frac{d}{{dx}}\,\,\left( {\mathop \smallint \limits_{f\left( x \right)}^{g\left( x \right)} \,\,\phi \,\,\left( t \right)\,\,dt} \right)$ is equal to

$\phi \,\,\left( {\,g\left( x \right)} \right)\,\, - \,\,\phi \,\,\left( {\,f\left( x \right)} \right)$

$\frac{1}{2}\,\,{\left[ {\,\phi \left( {\,g\,\,\left( x \right)} \right)} \right]^2}\,\, - \,\,\frac{1}{2}\,\,{\left[ {\,\phi \left( {\,f\,\,\left( x \right)} \right)} \right]^2}$

$\phi '\,\,\left( {\,g\left( x \right)} \right)\,\,g'\,\,\left( x \right)\,\, - \,\,\phi '\,\,\left( {f\left( x \right)} \right)\,\,f'\,\,\left( x \right)$

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

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

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

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JEE • MATHEMATICS • DEFINITE INTEGRATION

Definite integration practice · Mixed difficulty

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541 Medium

541 of 995

Analyzing +4 | -1

Let

{I_1} = \int \limits_1^2 \,\,\frac{1}{{\sqrt {1\,\, + \,\,{x^2}} }}\,\,dx

and

{I_2} = \int \limits_1^2 \,\,\frac{1}{x}\,\,dx

. Then

View answer and explanation

Correct option: B

{I_2} > {I_1}

Option B (I2 > I1) is correct because for x in [1, 2], 1/√(1+x²) < 1/x, implying that the integral I2 is greater than I1.

542 Easy

542 of 995

Application +4 | -1

Area enclosed by the parabola

a y=3\left(a^2-x^2\right)

and

x

-axis is

View answer and explanation

Correct option: A

4 a^2 \mathrm{sq}

, unit

Option A (

4 a^2

sq, unit) is correct because it represents the area enclosed by the parabola and the x-axis, calculated by integrating the function between its roots.

543 Easy

543 of 995

Application +4 | -1

\int \nolimits_0^{\pi /2} \frac{{2\,\sqrt {\cos \,\,\theta } }}{{3\,(\sqrt {\sin \,\,\theta } \,\, + \,\,\sqrt {\cos \,\,\theta )} }}\,\,d\mathrm{\theta }

is equal to

View answer and explanation

Correct option: A

\frac{\pi }{6}

Option A (

\frac{\pi}{6}

) is correct because using the property of definite integrals and adding the two forms of the integral simplifies the expression, leading to the result.

544 Medium

544 of 995

Analyzing +4 | -1

The value of the integral

\int \limits_0^\pi \,\,\log \,\,\left( {1\,\, + \,\,\cos \,\,x} \right)\,\,dx

View answer and explanation

Correct option: B

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

Option B (

- \pi \log \\,2

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

545 Easy

545 of 995

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: D

\pi /4

Option D (

\pi /4

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

546 Easy

546 of 995

Application +4 | -1

The value of the integral

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

View answer and explanation

Correct option: C

\frac{\pi}{12}

Option C is correct because by using the property

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

, the integral simplifies to

\frac{b-a}{2}

547 Easy

547 of 995

Analyzing +4 | -1

The value of

\int_0^{2\pi } {\frac{{x{{\sin }^8}x}}{{{{\sin }^8}x + {{\cos }^8}x}}dx}

is equal to

View answer and explanation

Correct option: D

{\pi ^2}

Option D (

{\pi ^2}

) is correct because it is the value of the definite integral after applying properties of definite integrals to simplify it.

548 Medium

548 of 995

Analyzing +4 | -1

Let

f(x)=|x-2|

and

g(x)=f(f(x)), x \in[0,4]

. Then

\int\limits_{0}^3(g(x)-f(x)) d x

is equal to:

View answer and explanation

Correct option: A

Option A (1) is correct because evaluating the definite integral of g(x)-f(x) from 0 to 3 gives a result of 1.

549 Easy

549 of 995

Application +4 | -1

Area bounded by the curve

x y-3 x-2 y-10=0, x

-axis and the lines

x=3, x=4

View answer and explanation

Correct option: C

16 \log 2+3

Option C (

16 \log 2+3

) is correct because it represents the definite integral of the given function between the specified limits, calculated after proper simplification.

550 Easy

550 of 995

Application +4 | -1

The value of integral

\int \nolimits_0^4 \,\,\left| {\,x\,\, - \,\,1\,} \right|\,\,dx\;\,\,\mathrm{is}

View answer and explanation

Correct option: B

\mathrm{5}

Option B (

\mathrm{5}

) is correct because the integral of the absolute value function is evaluated by splitting the integral at the point where the function changes sign.

551 Easy

551 of 995

Analyzing +4 | -1

The value of

\int\limits_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{(2+3 \sin x)}{\sin x(1+\cos x)} d x

is equal to

View answer and explanation

Correct option: D

\frac{10}{3}-\sqrt{3}+\log _{e} \sqrt{3}

Option D (

\frac{10}{3}-\sqrt{3}+\log _{e} \sqrt{3}

) is correct because the integral is solved by partial fraction decomposition and trigonometric substitution.

552 Medium

552 of 995

Application +4 | -1

\int\limits_{1}^e \frac{1+\log _{10} x}{x} d x=

View answer and explanation

Correct option: C

\frac{1}{2} \log _{10} e+1

Option C is correct because the integral can be split into two parts and evaluated, resulting in

\frac{1}{2} \log_{10} e + 1

553 Easy

553 of 995

Application +4 | -1

The area, enclosed by the curves

y = \sin x + \cos x\mathrm{ }\,\,\mathrm{and}\,\,\mathrm{ }y = |\cos x - \sin x|

and the lines

x = 0,\,\,x = \frac{\pi }{2},

View answer and explanation

Correct option: A

2\sqrt 2 (\sqrt 2 - 1)

Option A (

2\sqrt 2 (\sqrt 2 - 1)

) is correct because it represents the area between the curves, calculated by integrating the difference of the functions over the given interval.

554 Hard

554 of 995

Application +4 | -1

\int \limits_1^{\frac{{4\sqrt 3 }}{5} - 1} \,\,\frac{{x + 2}}{{\sqrt {{x^2} + 2x - 3} }}\,\,dx

equals to

View answer and explanation

Correct option: B

\frac{{2\,\sqrt 3 }}{3}\,\, + \,\,\frac{1}{2}\,\,\log \,\,3

Option B (

\frac{{2\,\sqrt 3 }}{3}\,\,+\,\,\frac{1}{2}\,\,\log \,3

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

555 Easy

555 of 995

Application +4 | -1

f

and

g

are continuous functions on

[0, \pi]

satisfying

f(x)+f(\pi-x)=1=g(x)+g(\pi-x), \quad

then

\int_0^\pi\{f(x)+g(x)\} d x

is equal to

View answer and explanation

Correct option: A

\pi

Option A (

\pi

) is correct because it uses the given functional equations to simplify the integral.

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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 Easy

556 of 995

Application +4 | -1

\frac{d}{{dx}}\,\,\left( {\int \limits_{f\left( x \right)}^{g\left( x \right)} \,\,\phi \,\,\left( t \right)\,\,dt} \right)

is equal to

View answer and explanation

Correct option: C

g'\left( x \right)\,\,\phi \left( {\,g\,\,\left( x \right)} \right)\,\, - \,\,f'\left( x \right)\,\,\phi \left( {\,f\,\,\left( x \right)} \right)

Option C (

g'(x)\,\,\phi (\,g\,\,( x ) )\, - \,f'(x)\,\,\phi (\,f\,\,( x ) )

) is correct because it directly applies the Leibniz rule for differentiating an integral with variable limits.

557 Easy

557 of 995

Application +4 | -1

Value of

x

satisfying

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

View answer and explanation

Correct option: C

2 \log 2

Option C (

2 \log 2

) is correct because solving the integral equation

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

leads to

x = \log 4 = 2 \log 2

558 Easy

558 of 995

Application +4 | -1

If [.] stands for the greatest integer function, then

\int\limits_{1}^2[3 x] d x

is equal to

View answer and explanation

Correct option: B

Option B (4) is correct because the integral is evaluated by breaking it into intervals where the greatest integer function has a constant value.

559 Medium

559 of 995

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.

560 Easy

560 of 995

Analyzing +4 | -1

I_1=\int\limits_{e}^{e^2} \frac{d x}{\log x}

and

I_2=\int\limits_{1}^2 \frac{e^x}{x} d x

, then

View answer and explanation

Correct option: A

I_1=I_2

Option A (

I_1=I_2

) is correct because by substituting

\log x = t

I_1

, we obtain the same integral as

I_2

561 Easy

561 of 995

Analyzing +4 | -1

{I_n} = \int \nolimits_0^{\pi /4} \,\,{\tan ^n}\,\,x\,\,dx

, then

\lim\limits_{n \to \infty } n\,\,\left[ {{I_n}\,\, + \,\,{I_{n + 2}}} \right]

is equal to

View answer and explanation

Correct option: B

\mathrm{1}

Option B (

\mathrm{1}

) is correct because the limit is evaluated by simplifying the integral and applying the limit definition.

562 Easy

562 of 995

Application +4 | -1

The value of

\int \limits_{1/e}^e \,\,\frac{{|\log \,\,x|}}{{{x^2}}}\,\,dx

, is

View answer and explanation

Correct option: C

2\,\,\left( {1\,\, - \,\,\frac{1}{e}} \right)

Option C,

2\,\,\left( {1\,\, - \,\,\frac{1}{e}} \right)

, is the correct answer because it is the result of evaluating the definite integral with the absolute value of the logarithm.

563 Easy

563 of 995

Application +4 | -1

The value of the integral

\int \limits_{ - 1}^1 \,\,\left[ {{x^2} + \left\{ x \right\}} \right]\,\,dx

, where

\left[ \cdot \right]

and

\left\{ \cdot \right\}

denote respectively the greatest integer function and fractional part function, is equal to

View answer and explanation

Correct option: B

\frac{{5 - \sqrt 5 }}{2}

Option B (

\frac{{5 - \sqrt 5 }}{2}

) is correct because evaluating the integral of the greatest integer function and fractional part function over the given limits results in this value.

564 Medium

564 of 995

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}

565 Easy

565 of 995

Analyzing +4 | -1

\int\limits_{0}^\pi[\cot x] d x,

[.] denotes the greatest integer function, is equal to

View answer and explanation

Correct option: D

-\frac{\pi}{2}

Option D (

-\frac{\pi}{2}

) is correct because the integral evaluates to this value after using the property of definite integrals and the greatest integer function.

566 Medium

566 of 995

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

567 Easy

567 of 995

Application +4 | -1

Let

I_n=\int\limits_{0}^{\pi / 2} x^n \cos x \mathrm{~d} x

then

I_8+56 I_6

is equal to

View answer and explanation

Correct option: B

\left(\frac{\pi}{2}\right)^8

Option B ((

\frac{π}{2}

)⁸) is correct because it uses integration by parts to find a recursive relationship for I_n and then simplifies the expression I_8 + 56I_6.

568 Easy

568 of 995

Application +4 | -1

Let

f: R \in R

be a continuous function given by

f(x+y)=f(x)+f(y)

for all

x, y \in R

.
If

\int_0^2 f(x) d x=\alpha

, then

\int_{-2}^2 f(x) d x

is equal to

View answer and explanation

Correct option: C

The correct answer is C (0) because the given functional equation implies that f(x) is of the form

\lambda x

, making it an odd function, and the integral of an odd function over a symmetric interval is zero.

569 Medium

569 of 995

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.

570 Easy

570 of 995

Analyzing +4 | -1

The value of the integral

\int\limits_{0}^1 e^{x^2} d x

lies in the interval

View answer and explanation

Correct option: C

(1, \mathrm{e})

Option C ((1, e)) is correct because the integral is bounded by the minimum and maximum values of the function within the given interval.

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