The value of $\mathop \smallint \nolimits_0^1 \,\,\frac{{{x^4} + \,\,1}}{{{x^2} + \,\,1}}\,\,dx$ is

$\frac{1}{6}\,\,\left( {3\,\, - \,\,4\pi } \right)$

$\frac{1}{6}\,\,\left( {3\,\, + \,\,4\,\pi } \right)$

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JEE • MATHEMATICS • DEFINITE INTEGRATION

Definite integration practice · Easy

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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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661 Easy

661 of 791

Application +4 | -1

Let

f(x)

be a continuous function on

[0,4]

satisfying

f(x) f(4-x)=1

. The value of the definite integral

\int_0^4 \frac{1}{1+f(x)} d x

equals

View answer and explanation

Correct option: C

Option C (2) is correct because it utilizes the substitution property of definite integrals and the given functional equation to simplify and solve for the integral's value.

662 Easy

662 of 791

Application +4 | -1

f\,\,\left( x \right) = \,\,\left\{ {\,\begin{array}{cccccccccccccccccccc} {\left| {\,x\,} \right|,}\\ {\left| {x - 2} \right|,} \end{array}\begin{array}{cccccccccccccccccccc} { - 1\,\,\, \le \,\,\,x\,\,\, \le 1}\\ {1\,\, < \,\,\,x\,\,\, \le 3} \end{array}} \right.

then

\int \nolimits_{ - 1}^3 \,\,f\,\,\left( x \right)\,\,dx

is equal to

View answer and explanation

Correct option: C

\mathrm{2}

Option D (2) is correct because the integral is evaluated by splitting the integral based on the piecewise function definition and then integrating each part.

663 Easy

663 of 791

Application +4 | -1

Area lying in the first quadrant and bounded by the curve

y=x^3

and the line

y=4 x

View answer and explanation

Correct option: D

The correct answer is D (8) because it represents the area between the curve and the line in the first quadrant, calculated by integrating the difference of the functions.

664 Easy

664 of 791

Application +4 | -1

Evaluate

\mathrm{I}=\int_0^1 \frac{\mathrm{x}^2-1}{\ln \mathrm{x}} \mathrm{dx}

View answer and explanation

Correct option: C

\ln 3

Option C (

\ln 3

) is correct because the integral is evaluated using Feynman's trick (differentiation under the integral sign).

665 Easy

665 of 791

Analyzing +4 | -1

Let

f\left( x \right) = \int \,\,\frac{{{x^2}\,dx}}{{\left( {1 + {x^2}} \right)\,\,\left( {1 + \sqrt {1 + {x^2}} } \right)}}

and

f\left( 0 \right) = 0

. Then,

f\left( 1 \right)

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because after solving the integral and applying the given condition

f(0) = 0

, we find that

f(1) = \log \\,\left( {1\\ + \\,\sqrt 2 } \right)\\,\\ - \\,\\,\frac{\pi }{4}

666 Easy

666 of 791

Application +4 | -1

I = \int \nolimits_{ - \pi }^\pi \,\,\frac{{{e^{\sin x}}}}{{{e^{\sin x\;}} + \,\,{e^{ - \mathrm{sinx}}}}}\,\,dx

, then I equals

View answer and explanation

Correct option: C

\pi

Option C (

\pi

) is correct because using the property of definite integrals and adding the original integral to the transformed integral simplifies the expression to

\pi

667 Easy

667 of 791

Analyzing +4 | -1

I_n \int_{-\pi}^\pi \frac{\sin n x}{\left(1+\pi^x\right) \sin x} d x, n=0,1,2, \ldots

then which one of the following is not true?

View answer and explanation

Correct option: D

I_n=I_{n+1}

Option D (

I_n=I_{n+1}

) is not true because

I_n

depends on whether n is even or odd, and

I_n = I_{n+2}

, not

I_{n+1}

668 Easy

668 of 791

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 absolute value can be removed by considering the interval and then integrating the resulting expression.

669 Easy

669 of 791

Application +4 | -1

\lim \limits_{{n \rightarrow \infty}} \sum\limits_{{r=1}}^n \frac{1}{n} e^{r / n}

, is

View answer and explanation

Correct option: B

e-1

Option B (

e-1

) is correct because the limit of the Riemann sum is equivalent to a definite integral.

670 Easy

670 of 791

Application +4 | -1

Let

f: R \rightarrow R

be a function defined by

f(x)=\left\{\begin{array}{r}{[x], x \leq 2} \\ 0, x > 2\end{array}\right.

, where

[x]

is the greatest integer less than or equal to

x

. If

I=\int\limits_{{-1}}^2 \frac{x f\left(x^2\right)}{2+f(x+1)} d x

, then the value, is

View answer and explanation

Correct option: A

\frac{1}{4}

Option A (

\frac{1}{4}

) is correct because the integral is evaluated by splitting the integration range and using the definition of the greatest integer function.

671 Easy

671 of 791

Application +4 | -1

The expression

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

, where

[x]

and

\{x\}

are integral and fractional parts of

x

and

n \in N

is equal to

View answer and explanation

Correct option: D

n-1

Option D (

n-1

) is correct because the integral of the integer part divided by the integral of the fractional part simplifies to

n-1

672 Easy

672 of 791

Application +4 | -1

The area of the region bounded by

y=x-1

and

x=3-y^2

View answer and explanation

Correct option: C

\frac{9}{2}

Option C (9/2) is correct because it is the area calculated by integrating the difference between the two functions with respect to y between the intersection points.

673 Easy

673 of 791

Analyzing +4 | -1

\int\limits_0^5 \cos \left(\pi\left(x-\left[\frac{x}{2}\right]\right)\right) d x

, where

[t]

denotes greatest integer less than or equal to

t

, is equal to

View answer and explanation

Correct option: D

The correct answer is D (0) because the integral evaluates to zero after splitting the interval and simplifying the cosine function.

674 Easy

674 of 791

Analyzing +4 | -1

The value of the integral

\int_0^2 \frac{\log \left(x^2+2\right)}{(x+2)^2} d x

View answer and explanation

Correct option: D

\frac{\sqrt{2}}{3} \tan ^{-1} \sqrt{2}-\frac{5}{12} \log 2+\frac{1}{12} \log 3

Option D,

\frac{\sqrt{2}}{3} \tan ^{-1} \sqrt{2}-\frac{5}{12} \log 2+\frac{1}{12} \log 3

, is correct because it results from correctly applying integration by parts and partial fraction decomposition to the given integral.

675 Easy

675 of 791

Application +4 | -1

\int\limits_{0}^{2 \pi} e^{x / 2} \sin \left(\frac{x}{2}+\frac{\pi}{4}\right) d x=

View answer and explanation

Correct option: C

Option C (0) is correct because the integral simplifies to an expression that evaluates to zero at both limits of integration.

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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

676 Easy

676 of 791

Application +4 | -1

\int \nolimits_0^a \,\,f\left( {2a\,\, - \,\,x} \right)\,\,dx = m\;\,\,\mathrm{and}\,\,\int \nolimits_0^a \,\,f\left( x \right)dx = n

, then

\int \nolimits_0^{2a} \,\,f\left( x \right)\,\,dx

is equal to

View answer and explanation

Correct option: D

m\,\, + \,\,n

Option D (

m\\ + \\ n

) is correct because

\int_0^{2a} f(x) dx

can be split into two integrals, and using the given information, it equals

m\\ + \\ n

677 Easy

677 of 791

Application +4 | -1

Let

\frac{d}{d x}(F(x))=\frac{e^{\sin x}}{x}, x > 0

\int\limits_{1}^4 \frac{2 e^{\sin x^2}}{x} d x=F(k)-F(1)

, then one of the possible values of

k

, is

View answer and explanation

Correct option: C

Option C is correct. By using substitution and the given derivative, the integral simplifies to F(16) - F(1), thus k=16.

678 Easy

678 of 791

Application +4 | -1

The value of the integral

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

View answer and explanation

Correct option: C

-\frac{\pi}{2}

Option C (

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

) is correct. The integral is evaluated using the fundamental theorem of calculus and the properties of the arctangent function.

679 Easy

679 of 791

Application +4 | -1

The value of

\int \nolimits_0^1 \,\,\frac{{{x^4} + \,\,1}}{{{x^2} + \,\,1}}\,\,dx

View answer and explanation

Correct option: D

\frac{1}{6}\,\,\left( {3\pi \,\, - \,\,4\,} \right)

Option D (

\frac{1}{6}\\,,\left( {3\pi \\, - \\,4\} \right)

) is correct because after dividing and integrating, the result is

\frac{1}{6}\\,,\left( {3\pi \\, - \\,4\} \right)

680 Easy

680 of 791

Application +4 | -1

The value of :

\int\limits_{{\pi / 6}}^{5 \pi / 6} \sqrt{4-4 \sin ^2 t} \,d t

, is

View answer and explanation

Correct option: B

Option B (2) is correct because the integral of the given expression is calculated by splitting the integral based on the absolute value and cosine function's sign.

681 Easy

681 of 791

Application +4 | -1

\int_\pi^{5 \pi / 4} \frac{\sin 2 x}{\cos ^4 x+\sin ^4 x} d x

is equal to

View answer and explanation

Correct option: B

\frac{\pi}{4}

Option B (

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

) is correct because after substitution and integration, the definite integral evaluates to

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

682 Easy

682 of 791

Analyzing +4 | -1

{I_1} = \int \limits_{1 - k}^k \,\,x\,\,\sin \,\,\left\{ {x\left( {1 - x} \right)} \right\}\,\,dx

and

{I_2} = \int \limits_{1 - k}^k \,\,\sin \,\,\left\{ {x\left( {1 - x} \right)} \right\}\,\,dx

, then

View answer and explanation

Correct option: B

2\,{I_1} = {I_2}

Option B (

2\,{I_1} = {I_2}

) is correct because by using the property of definite integrals and algebraic manipulation, we can derive this relationship.

683 Easy

683 of 791

Application +4 | -1

The value of

\int \nolimits_0^\infty \,\,\frac{{dx}}{{\left( {{a^2} + {x^2}} \right)}}

is equal to

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because the integral evaluates to (1/a)arctan(x/a) from 0 to infinity, which simplifies to pi/(2a).

684 Easy

684 of 791

Application +4 | -1

f:R \to R

and

\mathrm{g}:R \to R

are one to one, real valued functions, then the value of the integral

\int \nolimits_{ - \pi }^\pi \,\,\,\left[ {f\left( x \right)\,\, + \,\,f\,\left( { - x} \right)} \right]\,\,\left[ {\mathrm{g}\,\,\left( x \right)\,\, - \,\,\mathrm{g}\,\,\left( { - x} \right)} \right]\,\,dx

View answer and explanation

Correct option: A

\mathrm{0}

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.

685 Easy

685 of 791

Application +4 | -1

\lim \limits_{{n \rightarrow \infty}} \frac{1}{n} \sum\limits_{{r=1}}^{2 n} \frac{r}{\sqrt{n^2+r^2}}

equals

View answer and explanation

Correct option: B

-1+\sqrt{5}

Option B (

-1+\sqrt{5}

) is correct because the limit of the Riemann sum is equivalent to a definite integral.

686 Easy

686 of 791

Analyzing +4 | -1

f\left( x \right)

is defined

\left[ { - \,2,\;\,2} \right]\mathrm{\;}\,\,\mathrm{by\;}\,\,f\left( x \right) = 4{x^2} - \,\,3x + 1

and

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

,
then

\int \nolimits_{ - 2}^2 \,\,\mathrm{g}\,\,\left( x \right)dx

is equal to

View answer and explanation

Correct option: C

\mathrm{0}

The correct answer is C (0) because g(x) is an odd function, and the integral of an odd function over a symmetric interval is zero.

687 Easy

687 of 791

Analyzing +4 | -1

f(x)

is an odd periodic function defined on the interval

[-T / 2, T / 2]

, where

T

is the period of

f(x)

. Then

\phi(x)=\int\limits_{a}^x f(t) d t

, is

View answer and explanation

Correct option: A

periodic with period

T

Option A (

\pi \log 2

) is correct because the integral simplifies to an expression involving known integrals of logarithmic trigonometric functions.

688 Easy

688 of 791

Analyzing +4 | -1

The value of the integral

\int \limits_{ - \pi /2}^{\pi /2} \log \,\,\,\left( {\frac{{a\,\, - \,\,\sin \,\,\theta }}{{a\,\, + \,\,\sin \,\,\theta }}} \right)\,\,d\,\,\theta ,\,\,a > 0

, is

View answer and explanation

Correct option: A

\mathrm{0}

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

689 Easy

689 of 791

Application +4 | -1

\int \nolimits_1^3 \,\,\frac{{\cos \,\,(\log \,\,x)}}{x}\;\,\,dx

is equal to

View answer and explanation

Correct option: C

\sin \,(\log \,\,3)\,

Option C (

\sin \,(\log \,\,3)\,

) is correct because the integral is solved using a simple substitution, recognizing that the derivative of log(x) is 1/x.

690 Easy

690 of 791

Application +4 | -1

{I_1} = \int \nolimits_0^{\pi /2} \,\,\sin \,\,x\,\,dx,\,\,{I_2} = \int \nolimits_0^{\pi /2} \,\,{\sin ^3}\,\,x\,\,dx

then

View answer and explanation

Correct option: A

{I_1}\,\,\, > \,\,\,{I_2}

Option A is correct because sin(x) is greater than sin³(x) in the interval [0, pi/2], therefore the integral of sin(x) is greater than the integral of sin³(x).

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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