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JEE • MATHEMATICS • INDEFINITE INTEGRATION

Indefinite integration practice · Easy

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631 Easy
631 of 746
Understanding +4 | -1
If ∫cosec   x dx=f(x)+constant,  then   f(x)\int \mathrm{cosec}\;\,x\,dx = f\left( x \right) + \mathrm{constant},\mathrm{\;then}\,\;f\left( x \right) is equal to

View answer and explanation
Correct option: B
log⁡  ∣tan⁡  (x/2)∣\log \,\,|\tan \,\,\left( {x/2} \right)|
Option B (log⁡ ∣tan⁡ (x/2)∣\log \,|\tan \,\left( {x/2} \right)|) is correct because the integral of cosec(x) is equal to log|tan(x/2)| + constant.
632 Easy
632 of 746
Analyzing +4 | -1
If I=∫x27(6x2+5x+4)(x2+x+1)6dx=f(x)+CI=\int x^{27}\left(6 x^2+5 x+4\right)\left(x^2+x+1\right)^6 d x=f(x)+C, then f(x)f(x) is equal to
View answer and explanation
Correct option: A
17(x6+x5+x4)7\frac{1}{7}\left(x^6+x^5+x^4\right)^7
Option D (12x2ex2ex4+C\frac{1}{2} x^2 e^{x^2} e^{x^4}+C) is correct because by using the substitution x2=tx^2=t and recognizing the pattern for integration, the integral simplifies to the given expression.
633 Easy
633 of 746
Application +4 | -1
If In=∫0π4tan⁡nθdθI_n=\int_0^{\frac{\pi}{4}} \tan n \theta d \theta, then I2020+I2022I_{2020}+I_{2022}
View answer and explanation
Correct option: D
12021\frac{1}{2021}
Option D (12021\frac{1}{2021}) is correct because the recurrence relation In+In−2=1n−1I_n + I_{n-2} = \frac{1}{n-1} applies to this integral, leading to I2020+I2022=12021I_{2020} + I_{2022} = \frac{1}{2021}.
634 Easy
634 of 746
Understanding +4 | -1
∫xx(1+ln⁡∣x∣)dx\int x^x(1+\ln |x|) d x is equal to
View answer and explanation
Correct option: C
xx+cx^x+c
Option C is correct because the integral of x^x(1 + ln|x|) dx is x^x + c, which can be verified by differentiation.
635 Easy
635 of 746
Application +4 | -1
∫sec⁡xcosec⁡xlog⁡tan⁡xdx\int \frac{\sec x \operatorname{cosec} x}{\log \tan x} d x
View answer and explanation
Correct option: C
log⁡(log⁡tan⁡x)+c\log (\log \tan x)+c
Option C (log⁡(log⁡tan⁡x)+c\log (\log \tan x)+c) is correct because substituting t=log⁡tan⁡xt = \log \tan x simplifies the integral to ∫dtt\int \frac{dt}{t}, which integrates to log⁡t+c\log t + c.
636 Easy
636 of 746
Analyzing +4 | -1
∫etan⁡−1x(1+x2)[(sec⁡−11+x2)2+cos⁡−1(1−x21+x2)]dx,\int \frac{e^{\tan ^{-1} x}}{\left(1+x^2\right)}\left[\left(\sec ^{-1} \sqrt{1+x^2}\right)^2+\cos ^{-1}\left(\frac{1-x^2}{1+x^2}\right)\right] d x, (x>0)(x>0) is equal to
View answer and explanation
Correct option: C
etan⁡−1x⋅(sec⁡−1(1+x2))2+Ce^{\tan ^{-1} x} \cdot\left(\sec ^{-1}\left(\sqrt{1+x^2}\right)\right)^2+C
Option C (etan⁡−1x⋅(sec⁡−1(1+x2))2+Ce^{\tan^{-1} x} \cdot (\sec^{-1}(\sqrt{{1 + x^2}}))^2 + C) is correct because the integral simplifies to this form after applying trigonometric identities and substitution.
637 Easy
637 of 746
Application +4 | -1
Correct evaluation of ∫x(x−2)(x−1)dx\int \frac{x}{(x-2)(x-1)} d x is
View answer and explanation
Correct option: A
log⁡e(x−2)2(x−1)+p\log _e \frac{(x-2)^2}{(x-1)}+p
Option A, log⁡e(x−2)2(x−1)+p\log _e \frac{(x-2)^2}{(x-1)}+p, is the correct evaluation because it results from partial fraction decomposition and integration of the given expression.
638 Easy
638 of 746
Application +4 | -1
The value of ∫dx(x−a)(b−x)\int \frac{d x}{\sqrt{(x-a)(b-x)}}, is
View answer and explanation
Correct option: A
2sin⁡−1x−ab−a+C2 \sin ^{-1} \sqrt{\frac{x-a}{b-a}}+C
Option A (2sin⁡−1x−ab−a+C2 \sin ^{-1} \sqrt{\frac{x-a}{b-a}}+C) is correct because it is the result of the integral after applying the trigonometric substitution x=acos⁡2θ+bsin⁡2θx=a \cos ^2 \theta+b \sin ^2 \theta.
639 Easy
639 of 746
Application +4 | -1
∫  cos⁡  {2  tan⁡−1  1−x1+x }  dx\int \,\,\cos \,\,\left\{ {2\,\,{{\tan }^{ - 1}}\,\,\sqrt {\frac{{1 - x}}{{1 + x}}\,} } \right\}\,\,dx is equal to
View answer and explanation
Correct option: D
x22+c\frac{{{x^2}}}{2} + c
Option D (x22+c\frac{{{x^2}}}{2} + c) is correct because after simplifying the trigonometric expression, the integral becomes ∫xdx\int x dx, which evaluates to x22+c\frac{x^2}{2} + c.
640 Easy
640 of 746
Application +4 | -1
∫  x2+  x  −6(x−2)  (x−1)  dx\int \,\,\frac{{{x^2} + \,\,x\,\, - 6}}{{\left( {x - 2} \right)\,\,\left( {x - 1} \right)}}\,\,dx
View answer and explanation
Correct option: D
x  +  4  log⁡  (x−1)  +  cx\,\, + \,\,4\,\,\log \,\,\left( {x - 1} \right)\,\, + \,\,c
Option D (x,+,4,log⁡,(x−1),+,,cx\\,\\ + \\,4\\,\log \\,\left( {x - 1} \right)\\,\\ + \\,\\,c) is correct because the integral simplifies to a rational function that can be easily integrated after partial fraction decomposition.
641 Easy
641 of 746
Application +4 | -1
∫(x+3)ex(x+4)2dx=\int \frac{(x+3) e^x}{(x+4)^2} d x=
View answer and explanation
Correct option: C
exx+4+c\frac{e^x}{x+4}+c
Option C (exx+4+c\frac{{e^x}}{{x+4}}+c) is correct because the integral can be simplified and solved using integration by parts or by recognizing the form ∫ex[f(x)+f′(x)]dx=exf(x)+c\int e^x [f(x) + f'(x)] dx = e^x f(x) + c.
642 Easy
642 of 746
Application +4 | -1
∫sin⁡θ+cos⁡θ(sin⁡2θ)dθ\int {\frac{{\sin \theta + \cos \theta }}{{\sqrt {\left( {\sin 2\theta } \right)} }}d\theta } is equal to
View answer and explanation
Correct option: B
sin⁡−1θ(sin⁡θ−cos⁡θ){\sin ^{ - 1}}\theta \left( {\sin \theta - \cos \theta } \right)
Option C (sin⁡−1(cos⁡θ−sin⁡θ){\sin ^{ - 1}}\left( {\cos \theta - \sin \theta } \right)) is incorrect. Option B (sin⁡−1(sin⁡θ−cos⁡θ){\sin ^{ - 1}}\left( {\sin \theta - \cos \theta } \right)) is correct because after substituting t=sin⁡θ−cos⁡θt = \sin \theta - \cos \theta, the integral simplifies to arcsin(t).
643 Easy
643 of 746
Application +4 | -1
∫3x29−16x6dx=\int \frac{3 x^2}{\sqrt{9-16 x^6}} d x=
View answer and explanation
Correct option: A
14sin⁡−1(4x33)+c\frac{1}{4} \sin ^{-1}\left(\frac{4 x^3}{3}\right)+c
Option A is correct because the integral can be solved using substitution, resulting in 14sin⁡−1(4x33)+c\frac{1}{4} \sin^{-1}(\frac{4x^3}{3}) + c.
644 Easy
644 of 746
Application +4 | -1
∫(x2−x+1x2+1)ecot⁡−1(x)dx=f(x)⋅ecot⁡−1(x)+C\int\left(\frac{x^2-x+1}{x^2+1}\right) e^{\cot^ {-1}(x)} d x=f(x) \cdot e^{\cot^{-1}(x)}+C
where CC is constant of integration. Then f(x)f(x) is equal to:
View answer and explanation
Correct option: C
xx
Option C (xx) is correct because integrating the given expression using substitution cot⁡−1x=t\cot^{-1}x = t leads to f(x)=xf(x) = x.
645 Easy
645 of 746
Application +4 | -1
∫sin⁡x+8cos⁡x4sin⁡x+6cos⁡xdx=\int \frac{\sin x+8 \cos x}{4 \sin x+6 \cos x} d x=
View answer and explanation
Correct option: A
x+12log⁡∣4sin⁡x+6cos⁡x∣+Cx+\frac{1}{2} \log |4 \sin x+6 \cos x|+C
Option C (x+2log⁡∣2sin⁡x+3cos⁡x∣+Cx+2 \log |2 \sin x+3 \cos x|+C) is correct because the integral is solved by expressing the numerator as a linear combination of the denominator and its derivative.

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SEQUENCE OF SERIES 1764 MCQs DEFINITE INTEGRATION 1443 MCQs MATRICES 1363 MCQs PROBABILITY 1146 MCQs COMPLEX NUMBERS 980 MCQs DIFFERENTIAL EQUATIONS 977 MCQs
646 Easy
646 of 746
Application +4 | -1
I=∫cos⁡{2tan⁡−11−x1+x}dxI=\int \cos \left\{2 \tan ^{-1} \sqrt{\frac{1-x}{1+x}}\right\} d x is equal to
View answer and explanation
Correct option: A
12x2+C\frac{1}{2} x^2+C
Option A is correct because the integral can be simplified using trigonometric identities and a substitution.
647 Easy
647 of 746
Application +4 | -1
∫cos⁡3xelog⁡(sin⁡x)dx\int \cos ^3 x e^{\log (\sin x)} d x is equal to
View answer and explanation
Correct option: B
−cos⁡4x4+c-\frac{\cos ^4 x}{4}+c
Option A (−cos⁡4x4+c-\frac{\cos ^4 x}{4}+c) is correct because after simplifying elog⁡(sin⁡x)e^{\log (\sin x)} to sin⁡x\sin x, the integral becomes ∫cos⁡3xsin⁡xdx\int \cos^3 x \sin x dx, which is solved by substituting u=cos⁡xu = \cos x.
648 Easy
648 of 746
Application +4 | -1
If f:R→Rf: \mathbb{R} \rightarrow \mathbb{R} is a function such that f(x)+2f(1x)=3x ∀x≠0f(x)+2 f\left(\frac{1}{x}\right)=3 x \,\forall x \neq 0 then ∫xf(x)dx\int x f(x) dx is equal to
View answer and explanation
Correct option: A
2x−13x3+c2x - \frac{1}{3}{x^3} + c
Option A (2x−13x3+c2x - \frac{1}{3}{x^3} + c) is correct because solving the functional equation for f(x)f(x) and then integrating xf(x)xf(x) yields this result.
649 Easy
649 of 746
Application +4 | -1
∫e−xcosec⁡2(2e−x+5)dx=\int e^{-x} \operatorname{cosec}^2\left(2 e^{-x}+5\right) d x=
View answer and explanation
Correct option: A
12cot⁡(2e−x+5)+c\frac{1}{2} \cot \left(2 e^{-x}+5\right)+c
Option A (12cot⁡(2e−x+5)+c\frac{1}{2} \cot \left(2 e^{-x}+5\right)+c) is correct because it is the result of integrating the given expression using substitution.
650 Easy
650 of 746
Application +4 | -1
∫x−1(x−3)(x−2)dx=\int \frac{x-1}{(x-3)(x-2)} d x=
View answer and explanation
Correct option: B
log⁡(x−3)2−log⁡(x−2)+c\log (x-3)^2-\log (x-2)+c
Option B (log⁡(x−3)2−log⁡(x−2)+c\log (x-3)^2-\log (x-2)+c) is correct because the integral is solved by partial fraction decomposition.
651 Easy
651 of 746
Application +4 | -1
Let n≥2n \ge 2 be a natural number and 0<θ<π2.0 < \theta < \frac{\pi }{2}. Then ∫(sin⁡nθ−sin⁡θ)1ncos⁡θsin⁡n+1θdθ\int \frac{{{{({{\sin }^n}\theta - \sin \theta )}^{\frac{1}{n}}}\cos \theta }}{{{{\sin }^{n + 1}}\theta }}d\theta is equal to (where C is a constant of integration)
View answer and explanation
Correct option: C
nn2−1(1−1sin⁡n−1θ)n+1n+C\frac{n}{{{n^2} - 1}}{\left( {1 - \frac{1}{{{{\sin }^{n - 1}}\theta }}} \right)^{\frac{{n + 1}}{n}}} + C
Option D (1356) is correct because it is the sum of all two-digit numbers that leave a remainder of 2 or 5 when divided by 7.
652 Easy
652 of 746
Application +4 | -1
The value of ∫1xn(1+xn)1/ndx,(n∈N)\int \frac{1}{x^n\left(1+x^n\right)^{1 / n}} d x,(n \in N), is
View answer and explanation
Correct option: A
11−n{1+1xn}1−1n+C\frac{1}{1-n}\left\{1+\frac{1}{x^n}\right\}^{1-\frac{1}{n}}+C
Option A is correct because the integral is solved using a substitution method, resulting in the given expression with a fractional exponent.
653 Easy
653 of 746
Application +4 | -1
If ∫excos⁡4x dx=Ae5x(sin⁡4x+54cos⁡4x)+C\int e^x \cos 4 x \mathrm{~d} x=A e^{5 x}\left(\sin 4 x+\frac{5}{4} \cos 4 x\right)+C then AA is equal to
View answer and explanation
Correct option: A
441\frac{4}{41}
Option A is correct because after integrating by parts twice and solving for the integral, the coefficient A is found to be 441\frac{4}{41}.
654 Easy
654 of 746
Application +4 | -1
∫log⁡(x+1)dx=\int \log (x+1) d x=
View answer and explanation
Correct option: A
(x+1)log⁡(x+1)−x+c(x+1) \log (x+1)-x+c
Option A ((x+1)log⁡(x+1)−x+c(x+1) \log (x+1)-x+c) is correct because it is the result of integrating log⁡(x+1)\log(x+1) using integration by parts.
655 Easy
655 of 746
Analyzing +4 | -1
∫dxcos⁡(2x)cos⁡(4x)\int \frac{d x}{\cos (2 x) \cos (4 x)} is equal to
View answer and explanation
Correct option: A
122log⁡∣1+2sin⁡2x1−2sin⁡2x∣−12(log⁡∣sec⁡2x−tan⁡2x∣)+C\frac{1}{2 \sqrt{2}} \log \left|\frac{1+\sqrt{2} \sin 2 x}{1-\sqrt{2} \sin 2 x}\right|-\frac{1}{2}(\log |\sec 2 x-\tan 2 x|)+C
Option A is correct because after using trigonometric identities and integrating, the result matches the given expression.
656 Easy
656 of 746
Analyzing +4 | -1
The value of ∫(1+x2)dx(1−x2)1+x2+x4\int \frac{\left(1+x^2\right) d x}{\left(1-x^2\right) \sqrt{1+x^2+x^4}}, is
View answer and explanation
Correct option: A
−123log⁡∣x4+x2+1−3xx4+x2+1+3x∣+C-\frac{1}{2 \sqrt{3}} \log \left|\frac{\sqrt{x^4+x^2+1}-\sqrt{3} x}{\sqrt{x^4+x^2+1}+\sqrt{3} x}\right|+C
Option A (−123log⁡∣x4+x2+1−3xx4+x2+1+3x∣+C-\frac{1}{2 \sqrt{3}} \log \left|\frac{\sqrt{x^4+x^2+1}-\sqrt{3} x}{\sqrt{x^4+x^2+1}+\sqrt{3} x}\right|+C) is correct because the integral simplifies to this logarithmic form after appropriate substitutions.
657 Easy
657 of 746
Analyzing +4 | -1
If ∫f(x)dx=f(x)\int f(x) d x=f(x), then ∫{f(x)}2dx\int\{f(x)\}^2 d x is equal to
View answer and explanation
Correct option: A
12{f(x)}2\frac{1}{2}\{f(x)\}^2
Option A (12f(x)2\frac{1}{2}{f(x)}^2) is correct because if ∫f(x)dx=f(x)\int f(x) dx = f(x), then f(x)=Cexf(x) = Ce^x, and ∫f(x)2dx=12f(x)2\int {f(x)}^2 dx = \frac{1}{2}{f(x)}^2.
658 Easy
658 of 746
Application +4 | -1
If ∫f(x)dx=f(x)\int f(x) d x=f(x), then ∫{f(x)}2dx\int\{f(x)\}^2 d x is equal to
View answer and explanation
Correct option: A
12{f(x)}2\frac{1}{2}\{f(x)\}^2
Option A (12{f(x)}2\frac{1}{2}\{f(x)\}^2) is correct because the integral of the square of the function is half the square of the function, given the initial condition.
659 Easy
659 of 746
Application +4 | -1
∫xa3−x3dx=\int \sqrt{\frac{x}{a^3-x^3}} d x=
View answer and explanation
Correct option: B
23sin⁡−1(xa)3/2+c\frac{2}{3} \sin ^{-1}\left(\frac{x}{a}\right)^{3 / 2}+c
Option B is correct because the integral evaluates to 23sin⁡−1(xa)3/2+c\frac{2}{3} \sin^{-1} \left(\frac{x}{a}\right)^{3/2} + c after applying the trigonometric substitution.
660 Easy
660 of 746
Application +4 | -1
The value of ∫dxx+x3\int \frac{\mathrm{d} x}{\sqrt{x}+\sqrt[3]{x}} is
View answer and explanation
Correct option: C
2x−3(x3)+6(x6)−6log⁡(x6+1)+C2 \sqrt{x}-3(\sqrt[3]{x})+6(\sqrt[6]{x})-6 \log (\sqrt[6]{x}+1)+C
Option C is correct because the integral can be solved by using a substitution to simplify the expression.
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