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Definite Integration

48 questions · Mathematics · JEE Advanced

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JEE Advanced
Mathematics
Definite Integration

Definite Integration

48 questions · Mathematics · JEE Advanced

If $α = \frac{1}{2} \int 2 \frac{t a n ^{- 1} x}{2 x ^{2} - 3 x + 2} d x$ then the value of $7 tan (\frac{2 α 7}{π})$ is . (Here, the inverse trigonometric function $tan^{- 1} x$ …2025 · Shift 2 · Q32 · Numerical
Let $f : [0, \frac{π}{2}] \to [0, 1]$ be the function defined by $f (x) = sin^{2} x$ and let $g : [0, \frac{π}{2}] \to [0, \infty)$ be the function defined by $g (x) = \frac{π x}{2} - x^{2}$ .The value of $2 0 \int \frac{π}{2} f (x) g (x) d x - 0 \int \frac{π}{2} g (x) d x$ …2024 · Shift 2 · Q33 · Numerical
Let $f : [0, \frac{π}{2}] \to [0, 1]$ be the function defined by $f (x) = sin^{2} x$ and let $g : [0, \frac{π}{2}] \to [0, \infty)$ be the function defined by $g (x) = \frac{π x}{2} - x^{2}$ .The value of $\frac{16}{π ^{3}} 0 \int \frac{π}{2} f (x) g (x) d x$ …2024 · Shift 2 · Q34 · Numerical
Let $f : (0, 1) \to R$ be the function defined as $f (x) = n$ if $x \in [\frac{1}{n + 1}, \frac{1}{n})$ where $n \in N$ . Let $g : (0, 1) \to R$ be a function such that $x^{2} \int x \frac{1 - t}{t} d t < g (x) < 2 x$ …2023 · Shift 1 · Q21 · MCQ
For $x \in R$ , let $tan^{- 1} (x) \in (- \frac{π}{2}, \frac{π}{2})$ . Then the minimum value of the function $f : R \to R$ defined by $f (x) = 0 \int x t a n^{- 1} x \frac{e ^{(t - c o s t)}}{1 + t ^{2023}} d t$ …2023 · Shift 2 · Q25 · Numerical
Consider the equation $\int_{1}^{e} \frac{( l o g _{e} x ) ^{1/2}}{x ( a - ( l o g _{e} x ) ^{3/2} ) ^{2}} d x = 1, a \in (- \infty, 0) \cup (1, \infty)$ Which of the following statements is/are…2022 · Shift 1 · Q27 · Multiple correct
The greatest integer less than or equal to $\int_{1}^{2} lo g_{2} (x^{3} + 1) d x + \int_{1}^{l o g_{2} 9} (2^{x} - 1)^{\frac{1}{3}} d x$ is .2022 · Shift 2 · Q21 · Numerical
Let $f : [- \frac{π}{2}, \frac{π}{2}] \to R$ be a continuous function such that $f (0) = 1$ and $\int_{0}^{\frac{π}{3}} f (t) d t = 0$ . Then which of the following statements is(are) TRUE?2021 · Shift 2 · Q22 · Multiple correct
Let $g_{i} : [\frac{π}{8}, \frac{3 π}{8}] \to R, i = 1, 2$ , and $f : [\frac{π}{8}, \frac{3 π}{8}] \to R$ be functions such that $g_{1} (x) = 1, g_{2} (x) = ∣4 x - π ∣$ and $f (x) = sin^{2} x$ , for all $x \in [\frac{π}{8}, \frac{3 π}{8}]$ …2021 · Shift 2 · Q30 · Numerical
Let $g_{i} : [\frac{π}{8}, \frac{3 π}{8}] \to R, i = 1, 2$ , and $f : [\frac{π}{8}, \frac{3 π}{8}] \to R$ be functions such that $g_{1} (x) = 1, g_{2} (x) = ∣4 x - π ∣$ and $f (x) = sin^{2} x$ , for all $x \in [\frac{π}{8}, \frac{3 π}{8}]$ …2021 · Shift 2 · Q31 · Numerical
Let $ψ_{1} : [0, \infty) \to R$ , $ψ_{2} : [0, \infty) \to R$ , f : (0, $\infty$ ) $\to$ R and g : [0, $\infty$ ) $\to$ R be functions such that f(0) = g(0) = 0, $ψ_{1} (x) = e^{- x} + x, x \geq 0$ , $ψ_{2} (x) = x^{2} - 2 x - 2 e^{- x} + 2, x \geq 0$ …2021 · Shift 2 · Q34 · MCQ
Let $ψ_{1} : [0, \infty) \to R$ , $ψ_{2} : [0, \infty) \to R$ , f : (0, $\infty$ ) $\to$ R and g : [0, $\infty$ ) $\to$ R be functions such that f(0) = g(0) = 0, $ψ_{1} (x) = e^{- x} + x, x \geq 0$ , $ψ_{2} (x) = x^{2} - 2 x - 2 e^{- x} + 2, x \geq 0$ …2021 · Shift 2 · Q35 · MCQ
For any real number x, let [ x ] denote the largest integer less than or equal to x. If $I = 0 \int 10 [\frac{10 x}{x + 1}] d x$ , then the value of 9I is .2021 · Shift 2 · Q38 · Numerical
Which of the following inequalities is/are TRUE?2020 · Shift 1 · Q30 · Multiple correct
Let b be a nonzero real number. Suppose f : R $\to$ R is a differentiable function such that f(0) = 1. If the derivative f' of f satisfies the equation $f^{'} (x) = \frac{f ( x )}{b ^{2} + x ^{2}}$ for all x $\in$ R, then which of the following…2020 · Shift 2 · Q25 · Multiple correct
Let $f : R \to R$ be a differentiable function such that its derivative f' is continuous and f( $π$ ) = $-$ 6. If $F : [0, π] \to R$ is defined by $F (x) = \int_{0}^{x} f (t) d t$ , and if $\int_{0}^{π} (f^{'} (x) + F (x)) cos x d x$ = 2 then the value…2020 · Shift 2 · Q35 · Numerical
If $I = \frac{2}{π} - π /4 \int π /4 \frac{d x}{( 1 + e ^{s i n x} ) ( 2 - c o s 2 x )}$ , then 27I2 equals .................2019 · Shift 1 · Q34 · Numerical
The value of the integral $0 \int π /2 \frac{3 c o s θ}{( c o s θ + s i n θ ) ^{5}} d θ$ equals ..............2019 · Shift 2 · Q31 · Numerical
The value of the integral $\int_{0}^{1/2} \frac{1 + 3}{( ( x + 1 ) ^{2} ( 1 - x ) ^{6} ) ^{1/4}} d x$ is ........2018 · Shift 2 · Q25 · Numerical
If $I = \sum o l imi t s_{k = 1}^{98} \int_{k}^{k + 1} \frac{k + 1}{x ( x + 1 )} d x$ , then2017 · Shift 2 · Q27 · Multiple correct
The total number of distinct $x \in [0, 1]$ for which $0 \int x \frac{t ^{2}}{1 + t ^{4}} d t = 2 x - 1$ 2016 · Shift 1 · Q22 · Numerical
Let $f (x) = n \to \infty lim (\frac{n ^{n} ( x + n ) ( x + \frac{n}{2} ) ... ( x + \frac{n}{n} )}{n ! ( x ^{2} + n ^{2} ) ( x ^{2} + \frac{n ^{2}}{4} ) .... ( x ^{2} + \frac{n ^{2}}{n ^{2}} )})^{\frac{x}{n}},$ …2016 · Shift 2 · Q25 · Multiple correct
The value of $- \frac{π}{2} \int \frac{π}{2} \frac{x ^{2} c o s x}{1 + e ^{x}} d x$ is equal to2016 · Shift 2 · Q27 · MCQ
Let $f : R \to R$ be a function defined by $f (x) = {[x], 0, x \leq 2 x > 2$ where $[x]$ is the greatest integer less than or…2015 · Shift 1 · Q30 · Numerical
Let $f (x) = 7 tan^{8} x + 7 tan^{6} x - 3 tan^{4} x - 3 tan^{2} x$ for all $x \in (- \frac{π}{2}, \frac{π}{2}) .$ Then the correct expression(s) is (are)2015 · Shift 2 · Q34 · Multiple correct
The option(s) with the values of a and $L$ that satisfy the following equation is (are) $\frac{0 \int 4 π e ^{t} ( s i n ^{6} a t + c o s ^{4} a t ) d t}{0 \int π e ^{t} ( s i n ^{6} a t + c o s ^{4} a t ) d t} = L ?$ …2015 · Shift 2 · Q35 · Multiple correct
Let $f^{'} (x) = \frac{192 x ^{3}}{2 + s i n ^{4} π x}$ for all $x \in R$ with $f (\frac{1}{2}) = 0$ . If $m \leq 1/2 \int 1 f (x) d x \leq M,$ then the possible values of $m$ and…2015 · Shift 2 · Q36 · MCQ
If $α = 0 \int 1 (e^{9 x + 3 t a n^{- 1} x}) (\frac{12 + 9 x ^{2}}{1 + x ^{2}}) d x$ where $tan^{- 1} x$ takes only principal values, then the value of $(lo g_{e} ∣ 1 + α ∣ - \frac{3 π}{4})$ …2015 · Shift 2 · Q39 · Numerical
Let $f : (0, \infty) \to R$ be given by $f (x)$ = $\frac{1}{x} \int x \frac{e ^{- (t + \frac{1}{t})}}{t} d t$ . Then2014 · Shift 1 · Q26 · Multiple correct
The value of $0 \int 1 4 x^{3} {\frac{d ^{2}}{d x ^{2}} (1 - x^{2})^{5}} d x$ is2014 · Shift 1 · Q28 · Numerical
Let a $\in$ R and f : R $\to$ R be given by f(x) = x5 $-$ 5x + a. Then,2014 · Shift 1 · Q37 · Multiple correct
The following integral $\frac{π}{4} \int \frac{π}{2} (2 csc x)^{17} d x$ is equal to2014 · Shift 2 · Q36 · MCQ
List - $I$ P. The number of polynomials $f (x)$ with non-negative integer coefficients of degree $f (0) = 0$ , satisfying $f (0) = 0$ and $\int_{0}^{1} f (x) d x = 1,$ is Q. The number of points in the interval $[- 13, 13]$ …2014 · Shift 2 · Q37 · MCQ
Given that for each $a \in (0, 1), h \to 0^{+} lim h \int 1 - h t^{- a} (1 - t)^{a - 1} d t$ exists. Let this limit be $g (a) .$ In addition, it is given that the…2014 · Shift 2 · Q38 · MCQ
Given that for each $a \in (0, 1), h \to 0^{+} lim h \int 1 - h t^{- a} (1 - t)^{a - 1} d t$ exists. Let this limit be $g (a) .$ In addition, it is given that the…2014 · Shift 2 · Q39 · MCQ
Let $f$ $: [\frac{1}{2}, 1] \to R$ (the set of all real number) be a positive, non-constant and differentiable function such that $f^{'} (x) < 2 f (x)$ and $f (\frac{1}{2}) = 1.$ …2013 · Shift 1 · Q35 · MCQ
The value of the integral $- π /2 \int π /2 (x^{2} + 1 n \frac{π + x}{π - x}) cos x d x$ is2012 · Shift 2 · Q27 · MCQ
The value of $ℓ n 2 \int ℓ n 3 \frac{x s i n x ^{2}}{s i n x ^{2} + s i n ( ℓ n 6 - x ^{2} )} d x$ is2011 · Shift 1 · Q34 · MCQ
For any real number $x,$ let $x .$ denote the largest integer less than or equal to $x .$ Let $f$ be a real valued function defined on the interval $[- 10, 10]$ by $f (x) = {x - [x] 1 + [x] - x i f [x] i s o dd, i f [x] i s e v e n$ …2010 · Shift 1 · Q46 · Numerical
The value of $0 \int 1 \frac{x ^{4} ( 1 - x ) ^{4}}{1 + x ^{2}} d x$ is (are)2010 · Shift 1 · Q48 · MCQ
The value of $x \to 0 lim \frac{1}{x ^{3}} 0 \int x \frac{t l n ( 1 + t )}{t ^{4} + 4} d t$ is2010 · Shift 1 · Q49 · MCQ
Let $f$ be a real-valued function defined on the interval $(- 1, 1)$ such that $e^{- x} f (x) = 2 + 0 \int x t^{4} + 1 d t,$ for all $x \in (- 1, 1)$ , and let $f^{- 1}$ be the inverse…2010 · Shift 2 · Q31 · MCQ
Let $f : R \to R$ be a continuous function which satisfies $f (x) = 0 \int x f (t) d t$ . Then, the value of $f (ln 5)$ is .2009 · Shift 2 · Q20 · Numerical
If $I_{n} = - π \int π \frac{s i n n x}{( 1 + π ^{x} ) s i n x} d x, n = 0, 1, 2,$ .... then2009 · Shift 2 · Q25 · Multiple correct
Consider the functions defined implicitly by the equation $y^{3} - 3 y + x = 0$ on various intervals in the real line. If $x \in (- \infty, - 2) \cup (2, \infty)$ , the equation implicitly defines a unique real valued differentiable function $y = f (x)$ . If $x \in (- 2, 2)$ …2008 · Shift 1 · Q29 · MCQ
Consider the function $f : (- \infty, \infty) \to (- \infty, \infty)$ defined by $f (x) = \frac{x ^{2} - a x + 1}{x ^{2} + a x + 1}, 0 < a < 2.$ Let $g (x) = 0 \int e^{x} \frac{f ^{'} ( t )}{1 + t ^{2}} d t .$ …2008 · Shift 2 · Q30 · MCQ
$x \to \frac{π}{4} lim \frac{2 \int s e c ^{2} x f ( t ) d t}{x ^{2} - \frac{π ^{2}}{16}}$ equal2007 · Shift 1 · Q27 · MCQ
Match the integrals in Column I with the values in Column II. Includes table2007 · Shift 1 · Q44 · MCQ