The area of the closed igure bounded by $x = - 1,\;x = 2$ and $y = \left\{ {\begin{array}{*{20}{c}} { - {x^2} + 2\;,\;x \le 1}\\ {2x - 1\;\;,\;\;x > 1} \end{array}} \right.$ and the $x$-axis is

$\frac{{16}}{3}\,{\rm{sq unit}}$

$\frac{{10}}{3}\,{\rm{sq unit}}$

$\frac{{13}}{3}\,{\rm{sq unit}}$

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JEE • MATHEMATICS • DEFINITE INTEGRATION

Definite integration practice · Mixed difficulty

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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 Easy

511 of 995

Application +4 | -1

The value of

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

is equal to

View answer and explanation

Correct option: D

\mathrm{0}

Option D (0) is correct because the integrand is an odd function, and the integral is evaluated over a symmetric interval [-π, π].

512 Easy

512 of 995

Application +4 | -1

For

x > 0

, if

f(x) = \int\limits_1^x {\frac{{{{\log }_e}t}}{{\left( {1 + t} \right)}}dt}

then

f(e) + f\left( {\frac{1}{e}} \right)\,

is equal to

View answer and explanation

Correct option: C

\frac{1}{2}

Option C (

\frac{1}{2}

) is correct because after evaluating the integrals and using substitution, the expression simplifies to 1/2.

513 Hard

513 of 995

Application +4 | -1

The value of the integral

\int_0^1 x \cot ^{-1}\left(1-x^2+x^4\right) d x

View answer and explanation

Correct option: A

\frac{\pi}{4}-\frac{1}{2} \log _{\mathrm{e}} 2

Option A (

\frac{{\pi}}{{4}}-\frac{{1}}{{2}} \log _{\mathrm{{e}}} 2

) is correct because the integral is solved by using trigonometric identities and substitution.

514 Easy

514 of 995

Application +4 | -1

\int \nolimits_0^a \,\,\sqrt {{a^2} - \,\,{x^2}\,} \;dx

is equal to

View answer and explanation

Correct option: D

\frac{1}{4}\,\,\pi {a^2}

The correct answer is (1/4)πa² because the integral represents the area of a quarter circle with radius a.

515 Medium

515 of 995

Application +4 | -1

f\left( x \right) = \int \limits_0^{{x^2}} \,\,\sqrt {1 + \,\,{t^2}} \,dt

, then

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

equals

View answer and explanation

Correct option: C

2x\,\sqrt {1\,\, + \,\,{x^4}}

Option C (

2x\,\sqrt {1\,\, + \,\,{{{x^4}}}}

) is correct because it is the result of applying the Leibniz rule for differentiating under the integral sign.

516 Medium

516 of 995

Application +4 | -1

\int\limits_{0}^{1 \sqrt{3}} \frac{d x}{\left(2 x^2+1\right) \sqrt{x^2+1}}=

View answer and explanation

Correct option: C

\operatorname{Tan}^{-1} 1 / 2

Option C (

\operatorname{Tan}^{-1} 1 / 2

) is correct because after applying the appropriate substitutions and evaluating, the integral results in

\operatorname{Tan}^{-1} 1 / 2

517 Easy

517 of 995

Application +4 | -1

\lim\limits_{n \to \infty } \,\,{\left( {\frac{1}{n}\,\, + \,\,\frac{1}{{n + 1}} + \,\,... + \,\,\frac{1}{{3\,n}}} \right)^{}}

is equal to

View answer and explanation

Correct option: B

\log \,\,3

Option B is correct because the limit of the sum can be expressed as a definite integral, which evaluates to log 3.

518 Medium

518 of 995

Analyzing +4 | -1

Let

{I_n} = \int \limits_0^{\pi /2} \,\,{\cos ^n}\,x\,\,\,\cos \,\,n\,\,\,x\,\,dx

. Then,

{I_n}\,\,:\,\,{I_{n + 1}}

is equal to

View answer and explanation

Correct option: C

\mathrm{2 : 1}

Option C (2 : 1) is correct because the ratio

{I_n}\,:\,{I_{n + 1}}

simplifies to 2:1 after applying trigonometric identities and integration by parts.

519 Easy

519 of 995

Analyzing +4 | -1

Let

f:R \to R

be a continuous function satisfying

f\left( x \right) + f\left( {x + k} \right) = n,

for all

x \in R

where

k > 0

and

n

is a positive integer. If

{l_1} = \int\limits_0^{4nk} {f\left( x \right)} dx

and

{l_2} = \int\limits_{ - k}^{3k} {f\left( x \right)} dx,

then

View answer and explanation

Correct option: C

{l_1} + n{l_2} = 4{n^2}k

Option D (

{l_1} + n{l_2} = 4{n^2}k

) is correct because it is derived from the periodicity of the function and the properties of definite integrals.

520 Medium

520 of 995

Application +4 | -1

The value of the definite integral

\int_0^{\pi / 2} \sin x \sin 2 x \sin 3 x d x

is equal to

View answer and explanation

Correct option: D

\frac{1}{6}

Option D is correct because using trigonometric identities and substitution, the integral simplifies to 1/6.

521 Easy

521 of 995

Application +4 | -1

Area bounded by the curve

y=x e^{x^2}, x

-axis and the ordinates

x=0, x=a

View answer and explanation

Correct option: B

\frac{e^{a^2}-1}{2}

sq. unit

Option B (

\frac{{e^{a^2}-1}}{{2}}

sq. unit) is correct because it represents the definite integral of the given function from 0 to a, which calculates the area under the curve.

522 Easy

522 of 995

Analyzing +4 | -1

I = \int \limits_{ - \pi }^\pi \,\,\frac{{{{\sin }^2}\,\,x}}{{1 + {a^x}}}\,\,dx,\,\,a > 0

, then

I

equals

View answer and explanation

Correct option: B

\frac{\pi }{2}

Option B is correct because by using the property of definite integrals, the integral simplifies to π/2.

523 Medium

523 of 995

Analyzing +4 | -1

\int \limits_{ - \pi /3}^{\pi /3} \,\,\left( {\frac{a}{3}\,\,\left| {\tan x} \right|\,\, + \,\,\frac{{b\,\,\tan \,\,x}}{{1 + \,\,\sec \,\,x}}} \right)\,\,dx = 0

where

a,\;\,\,b,\;\,\,c

are constants, then

c =

View answer and explanation

Correct option: C

- \,\,\frac{a}{\pi }\,\,\ln \,\,2

Option C (

- \,\frac{a}{\pi }\\,\ln \\,2

) is correct because after evaluating the definite integral and using properties of even and odd functions, we find that

c = - \\,\frac{a}{\pi }\\,\ln \\,2

524 Easy

524 of 995

Application +4 | -1

The area of the closed igure bounded by

x = - 1,\;x = 2

and

y = \left\{ {\begin{array}{cccccccccccccccccccc} { - {x^2} + 2\;,\;x \le 1}\\ {2x - 1\;\;,\;\;x > 1} \end{array}} \right.

and the

x

-axis is

View answer and explanation

Correct option: A

\frac{{16}}{3}\,\mathrm{sq unit}

Option A (

\frac{{16}}{3}\,\mathrm{sq unit}

) is correct because it represents the definite integral of the piecewise function over the given interval.

525 Easy

525 of 995

Application +4 | -1

f(y)=e^y, g(y)=y ; y > 0

and

F(t)=\int\limits_{0}^t f(t-y) g(y) d y

, then

View answer and explanation

Correct option: C

F(t)=e^t-(1+t)

Option A (

\frac{3}{4}(2)^{4 / 3}-\frac{3}{4}

) is correct because the limit of the sum is converted to a definite 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

526 Easy

526 of 995

Application +4 | -1

Let

f: R \rightarrow R

be a continuous function which satisfies

f(2 x)=3 f(x) \,\forall \,x \in R

. If

\int\limits_{0}^1 f(x) d x=1

, then find the value of

\frac{1}{2} \int\limits_{1}^2 f(x) d x

View answer and explanation

Correct option: A

2.5

Option A (2.5) is correct because using the given functional equation and integral property, we can find the value of the required integral.

527 Hard

527 of 995

Application +4 | -1

For any real number

x

, the value of

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

, is

View answer and explanation

Correct option: C

x[x]-\frac{1}{2}[x]([x]+1)

Option C (

x[x]-\frac{1}{2}[x]([x]+1)

) is correct because it represents the general form of the integral of the greatest integer function from 0 to x.

528 Medium

528 of 995

Application +4 | -1

The integral

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

is equal to

View answer and explanation

Correct option: B

\mathrm{2}

Option B (2) is correct because the absolute value simplifies to 1 within the given interval, and the integral of 1 over the interval [-1, 1] is 2.

529 Easy

529 of 995

Application +4 | -1

\int\limits_{0}^2\left|x^2+2 x-3\right| \,d x

is equal to

View answer and explanation

Correct option: A

Option A (4) is correct because the integral of the absolute value of the quadratic function is calculated by splitting the integral at the root within the interval.

530 Medium

530 of 995

Application +4 | -1

\int \nolimits_{ - 1}^1 \,\,\left| {1 - x} \right|\;\,\,dx

is equal to

View answer and explanation

Correct option: C

\mathrm{2}

Option C (2) is correct because the integral of |1-x| from -1 to 1 is calculated by recognizing that 1-x is positive over the entire interval.

531 Easy

531 of 995

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: C

\log \,\,4

Option C is correct because the integral evaluates to

\log 4

after a suitable substitution.

532 Easy

532 of 995

Application +4 | -1

f\left( x \right) = \left\{ {\begin{array}{cccccccccccccccccccc} {{e^{\cos x}}\,\,\sin \,\,x,}\\ {2,} \end{array}\,\,\begin{array}{cccccccccccccccccccc} {\left| {\,x\,} \right|\,\, \le \,\,2}\\ {\mathrm{otherwise}} \end{array},} \right.

then

\int \nolimits_{ - 2}^3 \,\,f\left( x \right)\,\,dx

is equal to

View answer and explanation

Correct option: C

\mathrm{2}

Option C (2) is correct because the integral can be split into two parts, where the first part is zero due to the odd function and the second part evaluates to 2.

533 Easy

533 of 995

Application +4 | -1

The area of the region (in sq. units), in the first quadrant bounded by the parabola

y=9 x^2

and the lines

x=0, y=1

and

y=4

, is:

View answer and explanation

Correct option: D

14 / 9

Option D (14 / 9) is correct because the area under the curve

y=9x^2

between y=1 and y=4 is 14/9.

534 Easy

534 of 995

Application +4 | -1

The value of

\int\limits_0^\pi {\frac{{{e^{\cos x}}\sin x}}{{(1 + {{\cos }^2}x)({e^{\cos x}} + {e^{ - \cos x}})}}dx}

is equal to:

View answer and explanation

Correct option: C

\frac{\pi }{4}\,

Option C (π/4) is correct because the integral can be simplified using substitution and properties of definite integrals.

535 Easy

535 of 995

Application +4 | -1

f(x)

and

g(x)

are two differentiable functions on

[0,2]

such that

f^{\prime \prime}(x)-g^{\prime \prime}(x)=0, f^{\prime}(1)=2 g^{\prime}(1)=4 f(2)=3 g(2)=9

then

f(x)-g(x)

x=3 / 2

View answer and explanation

Correct option: D

Option D (5) is correct because by integrating twice and using the given conditions, we can find the value of f(x) - g(x) at x=3/2.

536 Easy

536 of 995

Application +4 | -1

\int \nolimits_0^{\pi /2} \,\,\frac{{\cos \,\,x}}{{1\,\, + \,\,\sin \,\,x}}\,\,\;dx

is equal to

View answer and explanation

Correct option: A

\log \,\,2

Option A (

\log \,2

) is correct because the definite integral of

\frac{{\cos \,x}}{{1 + \sin \,x}}

from 0 to

\frac{\pi}{2}

\log 2

537 Medium

537 of 995

Application +4 | -1

The value of

\int \limits_0^{2\pi } \,\,\left| {\,\cos \,\,x\,\, - \,\,\sin \,\,x\,} \right|dx

, is

View answer and explanation

Correct option: D

4\,\sqrt 2

Option D (

4\sqrt 2

) is correct because after evaluating the integral of the absolute value of the difference between cosine and sine functions over the interval

[0, 2\pi]

, the result is

4\sqrt{2}

538 Medium

538 of 995

Analyzing +4 | -1

The

\lim \limits_{{n \rightarrow \infty}} S_n

S_n=\frac{1}{2 n}+\frac{1}{\sqrt{4 n^2-1}}+\frac{1}{\sqrt{4 n^2-4}}+\ldots+\frac{1}{\sqrt{3 n^2+2 n-1}}

View answer and explanation

Correct option: D

\pi / 6

Option D (

\pi / 6

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

\pi/6

539 Easy

539 of 995

Analyzing +4 | -1

The value of

\int \nolimits_0^{\pi /2} \,\,\frac{{\cos \,\,3x\,\, + \,\,1}}{{2\,\,\cos \,\,x\,\, - \,\,1}}\,\,dx

View answer and explanation

Correct option: B

\mathrm{1}

Option B (1) is correct because after applying trigonometric identities and simplifying, the integral evaluates to 1.

540 Easy

540 of 995

Application +4 | -1

\int \limits_0^{{\pi ^2}/4} \sin \,\,\sqrt x \,dx

is equals to

View answer and explanation

Correct option: C

\mathrm{2}

Option C,

\mathrm{2}

, is correct because after substitution and integration by parts, the definite integral evaluates to 2.

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