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Indefinite Integrals

83 questions · Mathematics · JEE Main

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JEE Main
Mathematics
Indefinite Integrals

Indefinite Integrals

83 questions · Mathematics · JEE Main

$Let f (x) = \int x^{3} 3 - x^{2} d x . If 5 f (2) = - 4, then f (1) is equal to$ 2025 · 3 Apr · Shift 1 · Q32 · MCQ
If $\int \frac{( 1 + x ^{2} + x ) ^{10}}{( 1 + x ^{2} - x ) ^{9}} d x = \frac{1}{m} ((1 + x^{2} + x)^{n} (n 1 + x^{2} - x)) + C$ where C is the…2025 · 4 Apr · Shift 2 · Q46 · Numerical
If $\int (\frac{1}{x} + \frac{1}{x ^{3}}) (23 3 x^{- 24} + x^{- 26}) d x = - \frac{α}{3 ( α + 1 )} (3 x^{β} + x^{γ})^{\frac{α + 1}{α}} + C, x > 0, (α, β, γ \in Z)$ ,…2025 · 7 Apr · Shift 2 · Q48 · Numerical
If $\int e^{x} (\frac{x s i n ^{- 1} x}{1 - x ^{2}} + \frac{s i n ^{- 1} x}{( 1 - x ^{2} ) ^{3/2}} + \frac{x}{1 - x ^{2}}) d x = g (x) + C$ , where C is the constant of integration, then $g (\frac{1}{2})$ …2025 · 22 Jan · Shift 2 · Q35 · MCQ
Let $I (x) = \int \frac{d x}{( x - 11 ) ^{\frac{11}{13}} ( x + 15 ) ^{\frac{15}{13}}}$ . If $I (37) - I (24) = \frac{1}{4} (\frac{1}{b ^{\frac{1}{13}}} - \frac{1}{c ^{\frac{1}{13}}}), b, c \in N$ …2025 · 23 Jan · Shift 1 · Q39 · MCQ
Let $\int x^{3} sin x d x = g (x) + C$ , where $C$ is the constant of integration. If $8 (g (\frac{π}{2}) + g^{'} (\frac{π}{2})) = α π^{3} + β π^{2} + γ, α, β, γ \in Z$ , then $α + β - γ$ …2025 · 23 Jan · Shift 2 · Q42 · MCQ
If $\int \frac{2 x ^{2} + 5 x + 9}{x ^{2} + x + 1} d x = x x^{2} + x + 1 + α x^{2} + x + 1 + β lo g_{e} x + \frac{1}{2} + x^{2} + x + 1 + C$ , where $C$ is the constant of integration, then $α + 2 β$ …2025 · 24 Jan · Shift 2 · Q47 · Numerical
If $f (x) = \int \frac{1}{x ^{1/4} ( 1 + x ^{1/4} )} d x, f (0) = - 6$ , then $f (1)$ is equal to :2025 · 28 Jan · Shift 2 · Q41 · MCQ
If $\int cosec^{5} x d x = α cot x cosec x (cosec^{2} x + \frac{3}{2}) + β lo g_{x} tan \frac{x}{2} + C$ where $α, β \in R$ and $C$ is…2024 · 4 Apr · Shift 2 · Q52 · Numerical
If $\int \frac{1}{a ^{2} s i n ^{2} x + b ^{2} c o s ^{2} x} d x = \frac{1}{12} tan^{- 1} (3 tan x) +$ constant, then the maximum value of $a sin x + b cos x$ , is :2024 · 6 Apr · Shift 2 · Q48 · MCQ
Let $I (x) = \int \frac{6}{s i n ^{2} x ( 1 - c o t x ) ^{2}} d x$ . If $I (0) = 3$ , then $I (\frac{π}{12})$ is equal to2024 · 8 Apr · Shift 1 · Q50 · MCQ
If $\int \frac{1}{5 ( x - 1 ) ^{4} ( x + 3 ) ^{6}} d x = A (\frac{α x - 1}{β x + 3})^{B} + C$ , where $C$ is the constant of integration, then the value of $α + β + 20 AB$ is …2024 · 8 Apr · Shift 2 · Q53 · Numerical
Let $\int \frac{2 - t a n x}{3 + t a n x} d x = \frac{1}{2} (α x + lo g_{e} ∣ β sin x + γ cos x ∣) + C$ , where $C$ is the constant of integration. Then $α + \frac{γ}{β}$ is equal to :2024 · 9 Apr · Shift 1 · Q41 · MCQ
$The integral \int \frac{( x ^{8} - x ^{2} ) d x}{( x ^{12} + 3 x ^{6} + 1 ) t a n ^{- 1} ( x ^{3} + \frac{1}{x ^{3}} )} is equal to :$ 2024 · 27 Jan · Shift 2 · Q47 · MCQ
For $x \in (- \frac{π}{2}, \frac{π}{2})$ , if $y (x) = \int \frac{cosec x + s i n x}{cosec x s e c x + t a n x s i n ^{2} x} d x$ , and $lim_{x \to (\frac{π}{2})^{-}} y (x) = 0$ then $y (\frac{π}{4})$ …2024 · 29 Jan · Shift 1 · Q43 · MCQ
If $\int \frac{s i n ^{\frac{3}{2}} x + c o s ^{\frac{3}{2}} x}{s i n ^{3} x c o s ^{3} x s i n ( x - θ )} d x = A cos θ tan x - sin θ + B cos θ - sin θ cot x + C$ , where $C$ is the integration constant, then $A B$ …2024 · 29 Jan · Shift 2 · Q33 · MCQ
Let $I (x) = \int \frac{x ^{2} ( x s e c ^{2} x + t a n x )}{( x t a n x + 1 ) ^{2}} d x$ . If $I (0) = 0$ , then $I (\frac{π}{4})$ is equal to :2023 · 6 Apr · Shift 1 · Q30 · MCQ
Let $I (x) = \int \frac{( x + 1 )}{x ( 1 + x e ^{x} ) ^{2}} d x, x > 0$ . If $lim_{x \to \infty} I (x) = 0$ , then $I (1)$ is equal to :2023 · 8 Apr · Shift 1 · Q32 · MCQ
The integral $\int [(\frac{x}{2})^{x} + (\frac{2}{x})^{x}] ln (\frac{e x}{2}) d x$ is equal to :2023 · 8 Apr · Shift 2 · Q24 · MCQ
If $I (x) = \int e^{s i n^{2} x} (cos x sin 2 x - sin x) d x$ and $I (0) = 1$ , then $I (\frac{π}{3})$ is equal to :2023 · 10 Apr · Shift 1 · Q24 · MCQ
For $α, β, γ, δ \in N$ , if $\int ((\frac{x}{e})^{2 x} + (\frac{e}{x})^{2 x}) lo g_{e} x d x = \frac{1}{α} (\frac{x}{e})^{β x} - \frac{1}{γ} (\frac{e}{x})^{δ x} + C$ …2023 · 10 Apr · Shift 2 · Q27 · MCQ
Let $I (x) = \int \frac{x + 7}{x} d x$ and $I (9) = 12 + 7 lo g_{e} 7$ . If $I (1) = α + 7 lo g_{e} (1 + 22)$ , then $α^{4}$ is equal to .2023 · 12 Apr · Shift 1 · Q43 · Numerical
Let $f (x) = \int \frac{d x}{( 3 + 4 x ^{2} ) 4 - 3 x ^{2}}, ∣ x ∣0$ , then $α^{2} + β^{2}$ is equal to .2023 · 15 Apr · Shift 1 · Q41 · Numerical
Let $f (x) = \int \frac{2 x}{( x ^{2} + 1 ) ( x ^{2} + 3 )} d x$ . If $f (3) = \frac{1}{2} (lo g_{e} 5 - lo g_{e} 6)$ , then $f (4)$ is equal to2023 · 25 Jan · Shift 1 · Q30 · MCQ
If $\int sec 2 x - 1 d x = α lo g_{e} cos 2 x + β + cos 2 x (1 + cos \frac{1}{β} x) +$ constant, then $β - α$ is equal to .2023 · 30 Jan · Shift 2 · Q43 · Numerical
$The integral \int \frac{( 1 - \frac{1}{3} ) ( c o s x - s i n x )}{( 1 + \frac{2}{3} s i n 2 x )} d x is equal to$ 2022 · 26 Jul · Shift 2 · Q32 · MCQ
If $\int \frac{1}{x} \frac{1 - x}{1 + x} d x = g (x) + c$ , $g (1) = 0$ , then $g (\frac{1}{2})$ is equal to :2022 · 26 Jun · Shift 2 · Q31 · MCQ
If $\int \frac{( x ^{2} + 1 ) e ^{x}}{( x + 1 ) ^{2}} d x = f (x) e^{x} + C$ , where C is a constant, then $\frac{d ^{3} f}{d x ^{3}}$ at x = 1 is equal to :2022 · 27 Jun · Shift 1 · Q26 · MCQ
For $I (x) = \int \frac{s e c ^{2} x - 2022}{s i n ^{2022} x} d x$ , if $I (\frac{π}{4}) = 2^{1011}$ , then2022 · 29 Jul · Shift 2 · Q28 · MCQ
For real numbers $α$ , $β$ , $γ$ and $δ$ , if $\int \frac{( x ^{2} - 1 ) + t a n ^{- 1} ( \frac{x ^{2} + 1}{x} )}{( x ^{4} + 3 x ^{2} + 1 ) t a n ^{- 1} ( \frac{x ^{2} + 1}{x} )} d x = α lo g_{e} (tan^{- 1} (\frac{x ^{2} + 1}{x})) + β tan^{- 1} (\frac{γ ( x ^{2} + 1 )}{x}) + δ tan^{- 1} (\frac{x ^{2} + 1}{x}) + C$ …2021 · 16 Mar · Shift 2 · Q42 · Numerical
The integral $\int \frac{( 2 x - 1 ) c o s ( 2 x - 1 ) ^{2} + 5}{4 x ^{2} - 4 x + 6} d x$ is equal to (where c is a constant of integration)2021 · 18 Mar · Shift 1 · Q28 · MCQ
If $f (x) = \int \frac{5 x ^{8} + 7 x ^{6}}{( x ^{2} + 1 + 2 x ^{7} ) ^{2}} d x, (x \geq 0), f (0) = 0$ and $f (1) = \frac{1}{K}$ , then the value of K is2021 · 18 Mar · Shift 1 · Q45 · Numerical
If $\int \frac{c o s x - s i n x}{8 - s i n 2 x} d x = a sin^{- 1} (\frac{s i n x + c o s x}{b}) + c$ , where c is a constant of integration, then the ordered pair (a, b) is equal to :2021 · 24 Feb · Shift 1 · Q23 · MCQ
The value of the integral $\int \frac{s i n θ . s i n 2 θ ( s i n ^{6} θ + s i n ^{4} θ + s i n ^{2} θ ) 2 s i n ^{4} θ + 3 s i n ^{2} θ + 6}{1 - c o s 2 θ} d θ$ is :2021 · 25 Feb · Shift 1 · Q34 · MCQ
The integral $\int \frac{e ^{3 l o g_{e} 2 x} + 5 e ^{2 l o g_{e} 2 x}}{e ^{4 l o g_{e} x} + 5 e ^{3 l o g_{e} x} - 7 e ^{2 l o g_{e} x}} d x$ , x > 0, is equal to : (where c is a constant of integration)2021 · 25 Feb · Shift 2 · Q27 · MCQ
If $\int \frac{d x}{( x ^{2} + x + 1 ) ^{2}} = a tan^{- 1} (\frac{2 x + 1}{3}) + b (\frac{2 x + 1}{x ^{2} + x + 1}) + C$ , x > 0 where C is the constant of integration, then the value of…2021 · 27 Aug · Shift 1 · Q37 · Numerical
If $\int \frac{2 e ^{x} + 3 e ^{- x}}{4 e ^{x} + 7 e ^{- x}} d x = \frac{1}{14} (ux + v lo g_{e} (4 e^{x} + 7 e^{- x})) + C$ , where C is a constant of integration, then u + v is equal to .2021 · 27 Aug · Shift 2 · Q41 · Numerical
The integral $\int \frac{1}{4 ( x - 1 ) ^{3} ( x + 2 ) ^{5}} d x$ is equal to : (where C is a constant of integration)2021 · 31 Aug · Shift 1 · Q28 · MCQ
If $\int \frac{s i n x}{s i n ^{3} x + c o s ^{3} x} d x = α lo g_{e} ∣1 + tan x ∣ + β lo g_{e} ∣1 - tan x + tan^{2} x ∣ + γ tan^{- 1} (\frac{2 t a n x - 1}{3}) + C$ , when C is constant of…2021 · 31 Aug · Shift 2 · Q39 · Numerical
If $\int sin^{- 1} (\frac{x}{1 + x}) d x$ = A(x) $tan^{- 1} (x)$ + B(x) + C, where C is a constant of integration, then the ordered pair (A(x), B(x)) can be :2020 · 3 Sep · Shift 2 · Q38 · MCQ
Let $f (x) = \int \frac{x}{( 1 + x ) ^{2}} d x (x \geq 0)$ . Then f(3) – f(1) is eqaul to :2020 · 4 Sep · Shift 1 · Q34 · MCQ
The integral $\int (\frac{x}{x s i n x + c o s x})^{2} d x$ is equal to (where C is a constant of integration):2020 · 4 Sep · Shift 1 · Q36 · MCQ
If $\int (e^{2 x} + 2 e^{x} - e^{- x} - 1) e^{(e^{x} + e^{- x})} d x$ = $g (x) e^{(e^{x} + e^{- x})} + c$ where c is a constant of integration, then g(0) is equal to :2020 · 5 Sep · Shift 1 · Q24 · MCQ
If $\int \frac{c o s θ}{5 + 7 s i n θ - 2 c o s ^{2} θ} d θ$ = A $lo g_{e} ∣ B (θ) ∣ + C$ , where C is a constant of integration, then $\frac{B ( θ )}{A}$ can be :2020 · 5 Sep · Shift 2 · Q30 · MCQ
If $\int \frac{c o s x d x}{s i n ^{3} x ( 1 + s i n ^{6} x ) ^{2/3}} = f (x) (1 + sin^{6} x)^{1/ λ} + c$ where c is a constant of integration, then $λ f (\frac{π}{3})$ …2020 · 8 Jan · Shift 1 · Q27 · MCQ
If ƒ'(x) = tan–1(secx + tanx), $- \frac{π}{2} < x < \frac{π}{2}$ , and ƒ(0) = 0, then ƒ(1) is equal to :2020 · 9 Jan · Shift 1 · Q23 · MCQ
The integral $\int \frac{d x}{( x + 4 ) ^{\frac{8}{7}} ( x - 3 ) ^{\frac{6}{7}}}$ is equal to : (where C is a constant of integration)2020 · 9 Jan · Shift 1 · Q35 · MCQ
If $\int \frac{d θ}{c o s ^{2} θ ( t a n 2 θ + s e c 2 θ )} = λ tan θ + 2 lo g_{e} ∣ f (θ) ∣ + C$ where C is a constant of integration, then the ordered pair ( $λ$ …2020 · 9 Jan · Shift 2 · Q26 · MCQ
$\int \frac{s i n \frac{5 x}{2}}{s i n \frac{x}{2}} d x$ is equal to (where c is a constant of integration)2019 · 8 Apr · Shift 1 · Q41 · MCQ
If $\int \frac{d x}{x ^{3} ( 1 + x ^{6} ) ^{2/3}} = x f (x) (1 + x^{6})^{\frac{1}{3}} + C$ where C is a constant of integration, then the function ƒ(x) is equal to2019 · 8 Apr · Shift 2 · Q45 · MCQ
The integral $\int se c^{2/3} x cose c^{4/3} x dx$ is equal to (Hence C is a constant of integration)2019 · 9 Apr · Shift 1 · Q40 · MCQ
$\int e^{s e c x} (sec x tan x f (x) + sec x tan x + se x^{2} x) d x$ = esecxf(x) + C then a possible choice of f(x) is :-2019 · 9 Apr · Shift 2 · Q35 · MCQ
For x2 $e$ n $π$ + 1, n $\in$ N (the set of natural numbers), the integral $\int x \frac{2 s i n ( x ^{2} - 1 ) - s i n 2 ( x ^{2} - 1 )}{2 s i n ( x ^{2} - 1 ) + s i n 2 ( x ^{2} - 1 )} d x$ is equal to : (where c is a constant of integration)2019 · 9 Jan · Shift 1 · Q39 · MCQ
If $f (x) = \int \frac{5 x ^{8} + 7 x ^{6}}{( x ^{2} + 1 + 2 x ^{7} ) ^{2}} d x, (x \geq 0), f (0) = 0,$ then the value of $f (1)$ is :2019 · 9 Jan · Shift 2 · Q32 · MCQ
If $\int \frac{d x}{( x ^{2} - 2 x + 10 ) ^{2}} = A (tan^{- 1} (\frac{x - 1}{3}) + \frac{f ( x )}{x ^{2} - 2 x + 10}) + C$ where C is a constant of integration then :2019 · 10 Apr · Shift 1 · Q38 · MCQ
If $\int x^{5} e^{- x^{2}} d x = g (x) e^{- x^{2}} + c$ , where c is a constant of integration, then $g$ (–1) is equal to :2019 · 10 Apr · Shift 2 · Q28 · MCQ
Let n $\geq$ 2 be a natural number and $0 < θ < \frac{π}{2} .$ Then $\int \frac{( s i n ^{n} θ - s i n θ ) ^{1/ n} c o s θ}{s i n ^{n + 1} θ} d θ$ is equal to - (where C is a…2019 · 10 Jan · Shift 1 · Q26 · MCQ
If $\int$ x5.e $-$ 4x3 dx = $\frac{1}{48}$ e $-$ 4x3 f(x) + C, where C is a constant of inegration, then f(x) is equal to -2019 · 10 Jan · Shift 2 · Q43 · MCQ
If $\int \frac{1 - x ^{2}}{x ^{4}}$ dx = A(x) $(1 - x^{2})^{m}$ + C, for a suitable chosen integer m and a function A(x), where C is a constant of integration, then (A(x))m equals :2019 · 11 Jan · Shift 1 · Q32 · MCQ
If $\int \frac{x + 1}{2 x - 1} d x$ = f(x) $2 x - 1$ + C, where C is a constant of integration, then f(x) is equal to :2019 · 11 Jan · Shift 2 · Q28 · MCQ
The integral $\int \frac{2 x ^{3} - 1}{x ^{4} + x} d x$ is equal to : (Here C is a constant of integration)2019 · 12 Apr · Shift 1 · Q28 · MCQ
Let $a \in (0, \frac{π}{2})$ be fixed. If the integral $\int \frac{t a n x + t a n α}{t a n x - t a n α} d x$ = A(x) cos 2 $α$ + B(x) sin 2 $α$ + C, where C is a constant of integration, then the…2019 · 12 Apr · Shift 2 · Q26 · MCQ
The integral $\int$ cos(loge x) dx is equal to : (where C is a constant of integration)2019 · 12 Jan · Shift 1 · Q38 · MCQ
The integral $\int \frac{3 x ^{13} + 2 x ^{11}}{( 2 x ^{4} + 3 x ^{2} + 1 ) ^{4}} d x$ is equal to : (where C is a constant of integration)2019 · 12 Jan · Shift 2 · Q42 · MCQ
If $f (\frac{x - 4}{x + 2}) = 2 x + 1,$ (x $\in$ R $-$ {1, } $-$ 2}), then $\int f (x) d x$ is equal to : (where C is a constant of integration)2018 · 15 Apr · Shift 1 · Q44 · MCQ
If $\int \frac{2 x + 5}{7 - 6 x - x ^{2}} d x = A 7 - 6 x - x^{2} + B sin^{- 1} (\frac{x + 3}{4}) + C$ (where C is a constant of integration), then the ordered pair (A, B) is equal to :2018 · 15 Apr · Shift 2 · Q38 · MCQ
If $\int \frac{t a n x}{1 + t a n x + t a n ^{2} x} d x = x - \frac{K}{A} tan^{- 1} (\frac{K t a n x + 1}{A}) + C, (C$ is a constant of integration) then the ordered pair (K, A) is equal to :2018 · 16 Apr · Shift 1 · Q27 · MCQ
The integral $\int \frac{s i n ^{2} x c o s ^{2} x}{( s i n ^{5} x + c o s ^{3} x s i n ^{2} x + s i n ^{3} x c o s ^{2} x + c o s ^{5} x ) ^{2}} d x$ is equal to2018 · Shift 0 · Q39 · MCQ
The integral $\int 1 + 2 cot x (csc x + cot x) d x (0 < x < \frac{π}{2})$ is equal to : (where C is a constant of integration)2017 · 8 Apr · Shift 1 · Q48 · MCQ
If f $(\frac{3 x - 4}{3 x + 4})$ = x + 2, x $e - \frac{4}{3}$ , and $\int$ f(x) dx = A log $∣$ 1 $-$ x $∣$ + Bx + C, then the ordered pair (A, B) is equal to : (where C is a…2017 · 9 Apr · Shift 1 · Q41 · MCQ
Let $I_{n} = \int tan^{n} x d x, (n > 1) .$ If $I_{4} + I_{6}$ = $a tan^{5} x + b x^{5} + C$ , where C is a constant of integration, then the ordered pair $(a, b)$ is equal to2017 · Shift 0 · Q39 · MCQ
If $\int \frac{d x}{c o s ^{3} x 2 s i n 2 x} = (tan x)^{A} + C (tan x)^{B} + k,$ where k is a constant of integration, then A + B +C equals :2016 · 9 Apr · Shift 1 · Q43 · MCQ
The integral $\int \frac{d x}{( 1 + x ) x - x ^{2}}$ is equal to : (where C is a constant of integration.)2016 · 10 Apr · Shift 1 · Q39 · MCQ
The integral $\int \frac{2 x ^{12} + 5 x ^{9}}{( x ^{5} + x ^{3} + 1 ) ^{3}} d x$ is equal to :2016 · Shift 0 · Q27 · MCQ
The integral $\int \frac{d x}{x ^{2} ( x ^{4} + 1 ) ^{3/4}}$ equals :2015 · Shift 0 · Q31 · MCQ
The integral $\int (1 + x - \frac{1}{x}) e^{x + \frac{1}{x}} d x$ is equal to2014 · Shift 0 · Q30 · MCQ
If $\int f (x) d x = ψ (x),$ then $\int x^{5} f (x^{3}) d x$ is equal to2013 · Shift 0 · Q33 · MCQ
If the $\int \frac{5 t a n x}{t a n x - 2} d x = x + a ln ∣ sin x - 2 cos x ∣ + k,$ then $a$ is equal to :2012 · Shift 0 · Q33 · MCQ
The value of $2 \int \frac{s i n x d x}{s i n ( x - \frac{π}{4} )}$ is2008 · Shift 0 · Q37 · MCQ
$\int \frac{d x}{c o s x + 3 s i n x}$ equals2007 · Shift 0 · Q50 · MCQ
$\int {\frac{( l o g x - 1 )}{1 + ( l o g x ) ^{2}}}^{2} d x$ is equal to2005 · Shift 0 · Q78 · MCQ
If $\int \frac{s i n x}{s i n ( x - α )} d x = A x + B lo g sin (x - α), + C,$ then value of $(A, B)$ is2004 · Shift 0 · Q82 · MCQ
$\int \frac{d x}{c o s x - s i n x}$ is equal to2004 · Shift 0 · Q83 · MCQ