If area bounded by curve $f(x)$ and $x$-axis, $x=1$ to $x=b$ is $(b-1)$ $\sin (3 b+4)$, then $f(x)$ is

$3 x \cos (3 x+4)+\sin (3 x+4)$

If $f\,\,\left( x \right) = f\,\,\left( {a - x} \right)\,\,{\rm{and}}\,\,{\rm{g}}\left( x \right)\,\, + \,\,{\rm{g}}\left( {a - x} \right) = 2$,then the value of $\mathop \smallint \nolimits_0^a \,\,f\left( x \right)\,\,{\rm{g}}\,\,\left( x \right)\,\,dx\,\,{\rm{is}}$ is

$\mathop \smallint \nolimits_0^a \,\,f\,\,\left( x \right)\,\,dx$

$\mathop \smallint \nolimits_0^a \,\,g\,\,\left( x \right)\,\,dx$

$\mathop \smallint \nolimits_0^a \,\,[\,{\rm{g}}\left( x \right)\,\, - \,\,f\left( x \right)\,\,dx$

$\mathop \smallint \nolimits_0^a \,\,[\,{\rm{g}}\left( x \right)\,\, + \,\,f\left( x \right)\,\,dx$

The maximum value of $f(x)=\int_0^1 t \sin (x+\pi t) d t$ is

$\frac{1}{2 \pi^2} \sqrt{\pi^2+4}$

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JEE • MATHEMATICS • DEFINITE INTEGRATION

Definite integration practice · Easy

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Practice format and chapter quiz

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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Showing 601-630 of 791 questions

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601 Easy

601 of 791

Understanding +4 | -1

\int \nolimits_{ - 1}^1 \,\,{\sin ^3}\,\,x\,\,{\cos ^2}\,\,x\,dx

is equal to

View answer and explanation

Correct option: C

\mathrm{0}

Option C (0) is correct because the integrand is an odd function, and the integral is evaluated over a symmetric interval [-1, 1].

602 Easy

602 of 791

Application +4 | -1

\int \,\,{e^{ - \log x}}\,\,\;dx

is equal to

View answer and explanation

Correct option: D

\log \,\,\left| {\,x\,} \right|\,\, + \,\,c

Option D (

\log \,\,\left| {\,x\} \right|\,\, + \,\,c

) is correct because the integral of

e^{-\log x}

simplifies to the integral of 1/x, which is log|x| + c.

603 Easy

603 of 791

Application +4 | -1

The value of the integral

\int \nolimits_3^6 \,\,\frac{{\sqrt x }}{{\sqrt {9 - x} + \,\,\sqrt x }}\,\,dx

View answer and explanation

Correct option: A

\frac{3}{2}

Option A is correct because the integral evaluates to

\frac{3}{2}

after using the property of definite integrals.

604 Easy

604 of 791

Application +4 | -1

\int\limits_{{-\pi / 4}}^{\pi / 4} \frac{\sec ^2 x d x}{1+e^x}

equals

View answer and explanation

Correct option: D

Option D (1) is correct because the integral simplifies to 1 after using properties of even and odd functions and evaluating the remaining integral.

605 Easy

605 of 791

Application +4 | -1

If area bounded by curve

f(x)

and

x

-axis,

x=1

x=b

(b-1)

\sin (3 b+4)

, then

f(x)

View answer and explanation

Correct option: B

3(x-1) \cos (3 x+4)+\sin (3 x+4)

Option D (

3(x-1) \cos (3 x+4)+\sin (3 x+4)

) is correct because differentiating the area function gives the function f(x).

606 Easy

606 of 791

Application +4 | -1

f\,\,\left( x \right) = f\,\,\left( {a - x} \right)\,\,\mathrm{and}\,\,\mathrm{g}\left( x \right)\,\, + \,\,\mathrm{g}\left( {a - x} \right) = 2

,then the value of

\int \nolimits_0^a \,\,f\left( x \right)\,\,\mathrm{g}\,\,\left( x \right)\,\,dx\,\,\mathrm{is}

View answer and explanation

Correct option: A

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

Option A (

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

) is correct because by substituting x with a-x and using the given properties, the integral simplifies to this form.

607 Easy

607 of 791

Application +4 | -1

f(\alpha)=\int_1^\alpha \frac{\log _{10} t}{1+t} d t, \alpha>0

, then

f\left(e^3\right)+f\left(e^{-3}\right)

is equal to:

View answer and explanation

Correct option: D

\frac{9}{2 \log _e(10)}

Option D (

\frac{{9}}{{2 \log_e(10)}}

) is correct because after substituting and simplifying the integral, we arrive at this result.

608 Easy

608 of 791

Application +4 | -1

The value of

\int\limits_{0}^1 \frac{8 \log (1+x)}{1+x^2} d x

View answer and explanation

Correct option: D

\pi \log 2

Option D (

\pi \log 2

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

609 Easy

609 of 791

Application +4 | -1

\int\limits_{0}^{\infty} e^{-2 x}(\sin 2 x+\cos 2 x) d x=

View answer and explanation

Correct option: C

\frac{1}{2}

Option C (

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

) is correct. The integral is evaluated using integration by parts and the properties of exponential and trigonometric functions.

610 Easy

610 of 791

Analyzing +4 | -1

Let

f

be a real valued continuous function on

[0,1]

and

f(x)=x+\int_0^1(x-t) f(t) d t

. Then, which of the following points

(x, y)

lies on the curve

y=f(x)

View answer and explanation

Correct option: D

(6,8)

Option D ((6,8)) is correct because substituting x=6 into the derived function f(x) yields y=8, satisfying the equation.

611 Easy

611 of 791

Application +4 | -1

The value of

\int\limits_{0}^{\pi / 2} \frac{d x}{(\sqrt{\cos x}+\sqrt{\sin x})^4}

is equal to

View answer and explanation

Correct option: D

\frac{1}{3}

Option D (

\frac{1}{3}

) is correct because the integral is solved using trigonometric substitution and simplification, resulting in a value of

\frac{1}{3}

612 Easy

612 of 791

Application +4 | -1

The value of

\int\limits_0^{\pi /2} {\frac{{{{\sin }^3}x}}{{\sin x + \cos x}}dx} \,

View answer and explanation

Correct option: C

\frac{{\pi - 1}}{4}

Option C (

\frac{{\pi - 1}}{4}

) is correct because, by using the property of definite integrals and simplifying, the integral can be evaluated to this value.

613 Easy

613 of 791

Application +4 | -1

\int\limits_{0}^a \frac{d x}{x+\sqrt{a^2-x^2}}=

View answer and explanation

Correct option: B

\frac{\pi}{4}

Option B (

\frac{\pi}{4}

) is correct because the integral evaluates to

\frac{\pi}{4}

after substitution and simplification.

614 Easy

614 of 791

Understanding +4 | -1

The value of

\int \nolimits_{ - 2}^2 \,\,\left( {a{x^3}\, + \,\,bx\,\, + \,\,c} \right)dx

depends on the

View answer and explanation

Correct option: B

Value of

c

Option B (Value of

c

) is correct because the integral of odd functions over a symmetric interval is zero, leaving only the constant term's integral to depend on

c

615 Easy

615 of 791

Application +4 | -1

The value of

\alpha

(-\pi, 0)

satisfying

\sin \alpha+\int\limits_ \alpha^{2 \alpha} \cos 2 x \mathrm{~d} x=0

View answer and explanation

Correct option: C

-\pi / 3

Option C is correct because the equation can be solved by evaluating the integral and solving the resulting trigonometric equation.

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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

616 Easy

616 of 791

Analyzing +4 | -1

The value of the integral

\int \nolimits_0^{\pi /2} \,\,\log \,\,|\tan \,\,x|dx\,\,\mathrm{is}

View answer and explanation

Correct option: B

\mathrm{0}

Option B (0) is correct because the integral of log|tan x| from 0 to π/2 is 0 due to the symmetry of the function.

617 Easy

617 of 791

Analyzing +4 | -1

a f(x)+b f\left(\frac{1}{x}\right)=\frac{1}{x}-5, x \neq 0, a \neq b

, then

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

equals

View answer and explanation

Correct option: B

\frac{(\ln 2-5) a+\frac{7 b}{2}}{a^2-b^2}

The correct answer is A (

\frac{(\ln 2-5) a+\frac{7}{2} b}{a^2-b^2}

) because by solving the system of equations for

f(x)

and then integrating, the result matches this option.

618 Easy

618 of 791

Application +4 | -1

\int_0^{\pi / 4} \frac{\cos ^2 x \sin ^2 x}{\left(\cos ^3 x+\sin ^3 x\right)^2} d x

is equal to

View answer and explanation

Correct option: C

1 / 6

Option C (1/6) is correct because it is the result of evaluating the definite integral after a suitable substitution.

619 Easy

619 of 791

Analyzing +4 | -1

[t]

denotes the greatest integer

\leq t

, then the value of

\int_0^1\left[2 x-\left|3 x^2-5 x+2\right|+1\right] d x

View answer and explanation

Correct option: A

\frac{\sqrt{37}+\sqrt{13}-4}{6}

Option A (

\frac{\sqrt{37}+\sqrt{13}-4}{6}

) is correct because it is the value of the definite integral calculated by splitting the integral based on the absolute value function and evaluating each part.

620 Easy

620 of 791

Understanding +4 | -1

f

is integrable on

[a, b]

then

\int\limits_{a}^b\left[f(x)-f_{\text {ave }}\right] \mathrm{d} x

is equal to

View answer and explanation

Correct option: D

Option D (0) is correct because the integral of a function minus its average value over an interval is always zero.

621 Easy

621 of 791

Analyzing +4 | -1

The maximum value of

f(x)=\int_0^1 t \sin (x+\pi t) d t

View answer and explanation

Correct option: B

\frac{1}{\pi^2} \sqrt{\pi^2+4}

Option B,

\frac{1}{\pi^2} \sqrt{\pi^2+4}

, is correct because it is the maximum value obtained by evaluating the integral and then finding the maximum of the resulting trigonometric function.

622 Easy

622 of 791

Application +4 | -1

\int_1^4 \log _e[x] d x

equals

View answer and explanation

Correct option: A

\log _e 6

Option A (

\log _e 6

) is correct because the integral is evaluated by breaking it into intervals where the greatest integer function is constant.

623 Easy

623 of 791

Analyzing +4 | -1

Let

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

and

f(0)=0

. Then

f(1)

View answer and explanation

Correct option: B

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

The correct answer is B (

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

) because it is the result of evaluating the definite integral after finding the antiderivative and applying the initial condition f(0) = 0.

624 Easy

624 of 791

Application +4 | -1

f(0)=2, f^{\prime}(x)=f(x), \phi(x)=x+f(x)

then

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

View answer and explanation

Correct option: B

2 e^2

Option B (

2 e^2

) is correct because the integral can be solved by recognizing that f(x) = 2e^x and then using integration by parts.

625 Easy

625 of 791

Application +4 | -1

\int \nolimits_0^{2n\pi } \,\,\left\{ {|\sin x|\,\, - \,\,\,\left| {\,\frac{1}{2}\sin \,\,x\,} \right|} \right\}dx

equals

View answer and explanation

Correct option: B

2n

Option B (

2n

) is correct because the integral of the absolute value of sine over one period is 2, and there are 2n such half-periods in the interval [0, 2nπ].

626 Easy

626 of 791

Application +4 | -1

For any

n \in N

and

x \in {R^ + }

, the value of the integral

\int \limits_0^{n\left[ x \right]} \left( {x - \left[ x \right]} \right)dx

, is

View answer and explanation

Correct option: C

\frac{n}{2}\,\,\left[ x \right]

Option C (

\frac{n}{2}\,\,[ x ]

) is correct because the integral of the fractional part of x over one period is 1/2, and we have n[x] such periods.

627 Easy

627 of 791

Analyzing +4 | -1

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

is equal to

View answer and explanation

Correct option: D

\pi \int\limits_{0}^{\pi / 2} f(\cos x) d x

Option D (

\pi \int_{0}^{\pi / 2} f(\cos x) d x

) is correct because it is the result of applying the property of definite integrals and simplifying.

628 Easy

628 of 791

Application +4 | -1

The area of the region bounded by the curve

y=\sin x

and the x-axis in

[-\pi, \pi]

View answer and explanation

Correct option: A

Option A (4) is correct because it's the total area calculated by considering the symmetry of the sine function about the y-axis and integrating over the positive interval.

629 Easy

629 of 791

Application +4 | -1

The value of the integral

\int \limits_0^1 \,\,\frac{1}{{{{\left( {1 + {x^2}} \right)}^{3/2}}}}\,\,dx

, is

View answer and explanation

Correct option: B

1/\sqrt 2

Option B (

1/\sqrt 2

) is correct because it results from a trigonometric substitution and evaluating the resulting integral.

630 Easy

630 of 791

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: A

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

Option A (

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

) is correct because using the property of definite integrals,

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

, simplifies the integral.

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