Definite Integration

278 questions · Mathematics · JEE Main

Definite Integration

278 questions · Mathematics · JEE Main

Let [.] denote the greatest integer function. If $0 \int e^{3} [\frac{1}{e ^{x - 1}}] d x = α - lo g_{e} 2$ , then $α^{3}$ is equal to .2025 · 2 Apr · Shift 1 · Q50 · Numerical
Let $f : [1, \infty) \to [2, \infty)$ be a differentiable function. If $10 \int_{1}^{1} f (t) dt = 5 x f (x) - x^{5} - 9$ for all $x ⩾ 1$ , then the value of $f (3)$ is :2025 · 2 Apr · Shift 2 · Q27 · MCQ
Let $(a, b)$ be the point of intersection of the curve $x^{2} = 2 y$ and the straight line $y - 2 x - 6 = 0$ in the second quadrant. Then the integral $I = \int_{a}^{b} \frac{9 x ^{2}}{1 + 5 ^{x}} d x$ is equal to :2025 · 2 Apr · Shift 2 · Q40 · MCQ
$4 \int_{0}^{1} (\frac{1}{3 + x ^{2} + 1 + x ^{2}}) d x - 3 lo g_{e} (3)$ is equal to :2025 · 2 Apr · Shift 2 · Q43 · MCQ
Let the domain of the function $f (x) = lo g_{2} lo g_{4} lo g_{6} (3 + 4 x - x^{2})$ be $(a, b)$ . If $\int_{0}^{b - a} [x^{2}] d x = p - q - r, p, q, r \in N, gcd (p, q, r) = 1$ , where $[\cdot]$ is the…2025 · 3 Apr · Shift 1 · Q33 · MCQ
The integral $\int_{0}^{π} \frac{8 x d x}{4 c o s ^{2} x + s i n ^{2} x}$ is equal to2025 · 3 Apr · Shift 2 · Q38 · MCQ
The value of $\int_{- 1}^{1} \frac{( 1 + ∣ x ∣ - x ) e ^{x} + ( ∣ x ∣ - x ) e ^{- x}}{e ^{x} + e ^{- x}} d x$ is equal to2025 · 4 Apr · Shift 1 · Q34 · MCQ
Let $f (x) + 2 f (\frac{1}{x}) = x^{2} + 5$ and $2 g (x) - 3 g (\frac{1}{2}) = x, x > 0$ . If $α = \int_{1}^{2} f (x) d x$ , and $β = \int_{1}^{2} g (x) d x$ , then the value of $9 α + β$ is :2025 · 4 Apr · Shift 2 · Q42 · MCQ
The integral $\int_{0}^{π} \frac{( x + 3 ) s i n x}{1 + 3 c o s ^{2} x} d x$ is equal to2025 · 7 Apr · Shift 1 · Q39 · MCQ
The integral $- 1 \int \frac{3}{2} (∣ π^{2} x sin (π x)) d x$ is equal to:2025 · 8 Apr · Shift 2 · Q37 · MCQ
Let f(x) be a positive function and $I_{1} = - \frac{1}{2} \int 1 2 x f (2 x (1 - 2 x)) d x$ and $I_{2} = - 1 \int 2 f (x (1 - x)) d x$ . Then the value of $\frac{I _{2}}{I _{1}}$ is equal to 2025 · 8 Apr · Shift 2 · Q40 · MCQ
Let for $f (x) = 7 tan^{8} x + 7 tan^{6} x - 3 tan^{4} x - 3 tan^{2} x, I_{1} = \int_{0}^{π /4} f (x) d x$ and $I_{2} = \int_{0}^{π /4} x f (x) d x$ . Then $7 I_{1} + 12 I_{2}$ is equal to :2025 · 22 Jan · Shift 1 · Q39 · MCQ
The value of $\int_{e^{2}}^{e^{4}} \frac{1}{x} (\frac{e ^{((l o g_{e} x)^{2} + 1)^{- 1}}}{e ^{((l o g_{e} x)^{2} + 1)^{- 1}} + e ^{((6 - l o g_{e} x)^{2} + 1)^{- 1}}}) d x$ is2025 · 23 Jan · Shift 1 · Q36 · MCQ
If $I = \int_{0}^{\frac{π}{2}} \frac{s i n ^{\frac{3}{2}} x}{s i n ^{\frac{3}{2}} x + c o s ^{\frac{3}{2}} x} d x$ , then $\int_{0}^{2 I} \frac{x s i n x c o s x}{s i n ^{4} x + c o s ^{4} x} d x$ equals :2025 · 23 Jan · Shift 2 · Q26 · MCQ
If $I (m, n) = \int_{0}^{1} x^{m - 1} (1 - x)^{n - 1} d x, m, n > 0$ , then $I (9, 14) + I (10, 13)$ is2025 · 24 Jan · Shift 1 · Q31 · MCQ
If $\int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{96 x ^{2} c o s ^{2} x}{( 1 + e ^{x} )} d x = π (α π^{2} + β), α, β \in Z$ , then $(α + β)^{2}$ equals2025 · 28 Jan · Shift 1 · Q32 · MCQ
Let $f : R \to R$ be a twice differentiable function such that $f (2) = 1$ . If $F (x) = x f (x)$ for all $x \in R$ , $0 \int 2 x F^{'} (x) d x = 6$ …2025 · 28 Jan · Shift 2 · Q28 · MCQ
Let $f$ be a real valued continuous function defined on the positive real axis such that $g (x) = 0 \int x t f (t) d t$ . If $g (x^{3}) = x^{6} + x^{7}$ , then value of $r = 1 \sum 15 f (r^{3})$ is :2025 · 28 Jan · Shift 2 · Q43 · MCQ
The integral $80 0 \int \frac{π}{4} (\frac{s i n θ + c o s θ}{9 + 16 s i n 2 θ}) d θ$ is equal to :2025 · 29 Jan · Shift 1 · Q45 · MCQ
Let $f : (0, \infty) \to R$ be a twice differentiable function. If for some $a e 0, 0 \int 1 f (λ x) d λ = a f (x), f (1) = 1$ and $f (16) = \frac{1}{8}$ , then $16 - f^{'} (\frac{1}{16})$ …2025 · 29 Jan · Shift 1 · Q49 · Numerical
Let $f (x) = 0 \int x t (t^{2} - 9 t + 20) dt, 1 \leq x \leq 5$ . If the range of $f$ is $[α, β]$ , then $4 (α + β)$ equals :2025 · 29 Jan · Shift 2 · Q41 · MCQ
If $24 0 \int \frac{π}{4} [sin 4 x - \frac{π}{12} + [2 sin x]] d x = 2 π + α$ , where $[\cdot]$ denotes the greatest integer function, then $α$ is equal to .2025 · 29 Jan · Shift 2 · Q46 · Numerical
If $t \to 0 lim (0 \int 1 (3 x + 5)^{t} d x)^{\frac{1}{t}} = \frac{α}{5 e} (\frac{8}{5})^{\frac{2}{3}}$ , then $α$ is equal to .2025 · 29 Jan · Shift 2 · Q49 · Numerical
The value of the integral $0 \int π /4 \frac{x d x}{s i n ^{4} ( 2 x ) + c o s ^{4} ( 2 x )}$ equals :2024 · 1 Feb · Shift 1 · Q32 · MCQ
If $- π /2 \int π /2 \frac{8 2 c o s x d x}{( 1 + e ^{s i n x} ) ( 1 + s i n ^{4} x )} = α π + β lo g_{e} (3 + 22)$ , where $α, β$ are integers, then $α^{2} + β^{2}$ …2024 · 1 Feb · Shift 1 · Q58 · Numerical
If $0 \int \frac{π}{3} cos^{4} x d x = a π + b 3$ , where $a$ and $b$ are rational numbers, then $9 a + 8 b$ is equal to :2024 · 1 Feb · Shift 2 · Q36 · MCQ
The value of $0 \int 1 (2 x^{3} - 3 x^{2} - x + 1)^{\frac{1}{3}} d x$ is equal to :2024 · 1 Feb · Shift 2 · Q46 · MCQ
Let $f : (0, \infty) \to R$ and $F (x) = 0 \int x t f (t) dt$ . If $F (x^{2}) = x^{4} + x^{5}$ , then $r = 1 \sum 12 f (r^{2})$ is equal to …2024 · 1 Feb · Shift 2 · Q55 · Numerical
$Let f (x) = {- 2, x - 2, - 2 \leq x \leq 0 0 < x \leq 2 and h (x) = f (∣ x ∣) + ∣ f (x) ∣ . Then \int_{- 2}^{2} h (x) d x is equal to:$ …2024 · 4 Apr · Shift 1 · Q31 · MCQ
If the shortest distance between the lines $\frac{x + 2}{2} = \frac{y + 3}{3} = \frac{z - 5}{4}$ and $\frac{x - 3}{1} = \frac{y - 2}{- 3} = \frac{z + 4}{2}$ is $\frac{38}{3 5} k$ , and $0 \int k [x^{2}] d x = α - α$ …2024 · 4 Apr · Shift 1 · Q52 · Numerical
If $\int_{0}^{\frac{π}{4}} \frac{s i n ^{2} x}{1 + s i n x c o s x} d x = \frac{1}{a} lo g_{e} (\frac{a}{3}) + \frac{π}{b 3}$ , where $a, b \in N$ , then $a + b$ …2024 · 4 Apr · Shift 1 · Q56 · Numerical
Let $f (x) = \int_{0}^{x} (t + sin (1 - e^{t})) d t, x \in R$ . Then, $lim_{x \to 0} \frac{f ( x )}{x ^{3}}$ is equal to2024 · 4 Apr · Shift 2 · Q37 · MCQ
If the value of the integral $- 1 \int 1 \frac{c o s α x}{1 + 3 ^{x}} d x$ is $\frac{2}{π}$ .Then, a value of $α$ is2024 · 4 Apr · Shift 2 · Q46 · MCQ
The integral $0 \int π /4 \frac{136 s i n x}{3 s i n x + 5 c o s x} d x$ is equal to :2024 · 5 Apr · Shift 1 · Q41 · MCQ
The value of $\int_{- π}^{π} \frac{2 y ( 1 + s i n y )}{1 + c o s ^{2} y} d y$ is :2024 · 5 Apr · Shift 1 · Q46 · MCQ
Let $β (m, n) = 0 \int 1 x^{m - 1} (1 - x)^{n - 1} d x, m, n > 0$ . If $0 \int 1 (1 - x^{10})^{20} d x = a \times β (b, c)$ …2024 · 5 Apr · Shift 2 · Q41 · MCQ
If $f (t) = 0 \int π \frac{2 x d x}{1 - c o s ^{2} t s i n ^{2} x}, 0 < t < π$ , then the value of $0 \int \frac{π}{2} \frac{π ^{2} dt}{f ( t )}$ equals .2024 · 5 Apr · Shift 2 · Q57 · Numerical
$0 \int π /4 \frac{c o s ^{2} x s i n ^{2} x}{( c o s ^{3} x + s i n ^{3} x ) ^{2}} d x$ is equal to2024 · 6 Apr · Shift 1 · Q44 · MCQ
Let $r_{k} = \frac{\int _{0}^{1} ( 1 - x ^{7} ) ^{k} d x}{\int _{0}^{1} ( 1 - x ^{7} ) ^{k + 1} d x}, k \in N$ . Then the value of $\sum_{k = 1}^{10} \frac{1}{7 ( r _{k} - 1 )}$ is equal to .2024 · 6 Apr · Shift 1 · Q58 · Numerical
Let $[t]$ denote the largest integer less than or equal to $t$ . If $0 \int 3 ([x^{2}] + [\frac{x ^{2}}{2}]) d x = a + b 2 - 3 - 5 + c 6 - 7$ , where $a, b, c \in Z$ …2024 · 6 Apr · Shift 2 · Q52 · Numerical
The value of $k \in N$ for which the integral $I_{n} = \int_{0}^{1} (1 - x^{k})^{n} d x, n \in N$ , satisfies $147 I_{20} = 148 I_{21}$ is2024 · 8 Apr · Shift 1 · Q32 · MCQ
Let $α \int l o g_{e} 4 \frac{d x}{e ^{x} - 1} = \frac{π}{6}$ . Then $e^{α}$ and $e^{- α}$ are the roots of the equation :2024 · 8 Apr · Shift 2 · Q38 · MCQ
Let $lim_{n \to \infty} (\frac{n}{n ^{4} + 1} - \frac{2 n}{( n ^{2} + 1 ) n ^{4} + 1} + \frac{n}{n ^{4} + 16} - \frac{8 n}{( n ^{2} + 4 ) n ^{4} + 16} + \dots + \frac{n}{n ^{4} + n ^{4}} - \frac{2 n \cdot n ^{2}}{( n ^{2} + n ^{2} ) n ^{4} + n ^{4}})$ …2024 · 9 Apr · Shift 1 · Q52 · Numerical
The integral $\int_{1/4}^{3/4} cos (2 cot^{- 1} \frac{1 - x}{1 + x}) d x$ is equal to2024 · 9 Apr · Shift 2 · Q36 · MCQ
$lim_{x \to \frac{π}{2}} (\frac{\int _{x^{3}}^{(π /2)^{3}} ( s i n ( 2 t ^{1/3} ) + c o s ( t ^{1/3} ) ) d t}{( x - \frac{π}{2} ) ^{2}})$ is equal to2024 · 9 Apr · Shift 2 · Q40 · MCQ
The value of the integral $\int_{- 1}^{2} lo g_{e} (x + x^{2} + 1) d x$ is2024 · 9 Apr · Shift 2 · Q41 · MCQ
If $0 \int 1 \frac{1}{3 + x + 1 + x} d x = a + b 2 + c 3$ , where $a, b, c$ are rational numbers, then $2 a + 3 b - 4 c$ is equal…2024 · 27 Jan · Shift 1 · Q31 · MCQ
If $(a, b)$ be the orthocentre of the triangle whose vertices are $(1, 2), (2, 3)$ and $(3, 1)$ , and $I_{1} = a \int b x sin (4 x - x^{2}) d x, I_{2} = a \int b sin (4 x - x^{2}) d x$ …2024 · 27 Jan · Shift 1 · Q35 · MCQ
For $0 < a < 1$ , the value of the integral $0 \int π \frac{d x}{1 - 2 a c o s x + a ^{2}}$ is :2024 · 27 Jan · Shift 2 · Q49 · MCQ
Let $f (x) = 0 \int x g (t) lo g_{e} (\frac{1 - t}{1 + t}) dt$ , where $g$ is a continuous odd function. If $\int_{- π /2}^{π /2} (f (x) + \frac{x ^{2} c o s x}{1 + e ^{x}}) d x = (\frac{π}{α})^{2} - α$ …2024 · 27 Jan · Shift 2 · Q54 · Numerical
$x \to \frac{π}{2} lim \frac{1}{( x - \frac{π}{2} ) ^{2}} x^{3} \int (\frac{π}{2})^{3} cos (t^{\frac{1}{3}}) d t$ is equal to2024 · 29 Jan · Shift 1 · Q48 · MCQ
If the value of the integral $\int_{- \frac{π}{2}}^{\frac{π}{2}} (\frac{x ^{2} c o s x}{1 + π ^{x}} + \frac{1 + s i n ^{2} x}{1 + e ^{s i n x^{2123}}}) d x = \frac{π}{4} (π + a) - 2$ , then the value of $a$ is2024 · 29 Jan · Shift 1 · Q50 · MCQ
Let the slope of the line $45 x + 5 y + 3 = 0$ be $27 r_{1} + \frac{9 r _{2}}{2}$ for some $r_{1}, r_{2} \in R$ . Then $lim_{x \to 3} (\int_{3}^{x} \frac{8 t ^{2}}{\frac{3 r _{2} x}{2} - r _{2} x ^{2} - r _{1} x ^{3} - 3 x} d t)$ is equal to …2024 · 29 Jan · Shift 2 · Q54 · Numerical
If $\int_{\frac{π}{6}}^{\frac{π}{3}} 1 - sin 2 x d x = α + β 2 + γ 3$ , where $α, β$ and $γ$ are rational numbers, then $3 α + 4 β - γ$ is equal to .2024 · 29 Jan · Shift 2 · Q60 · Numerical
The value of $lim_{n \to \infty} \sum_{k = 1}^{n} \frac{n ^{3}}{( n ^{2} + k ^{2} ) ( n ^{2} + 3 k ^{2} )}$ is :2024 · 30 Jan · Shift 1 · Q32 · MCQ
Let $f : [- \frac{π}{2}, \frac{π}{2}] \to R$ be a differentiable function such that $f (0) = \frac{1}{2}$ . If the $$\lim_{x \rightarrow 0} \frac{x \int_0^x f(\mathrm{t})…2024 · 30 Jan · Shift 1 · Q40 · MCQ
The value of $9 0 \int 9 [\frac{10 x}{x + 1}] d x$ , where $[t]$ denotes the greatest integer less than or equal to $t$ , is2024 · 30 Jan · Shift 1 · Q53 · Numerical
Let $f : R \to R$ be a function defined by $f (x) = \frac{x}{( 1 + x ^{4} ) ^{1/4}}$ , and $g (x) = f (f (f (f (x))))$ . Then, $18 \int_{0}^{25} x^{2} g (x) d x$ is equal to2024 · 30 Jan · Shift 2 · Q31 · MCQ
Let $y = f (x)$ be a thrice differentiable function in $(- 5, 5)$ . Let the tangents to the curve $y = f (x)$ at $(1, f (1))$ and $(3, f (3))$ make angles $π /6$ and $π /4$ , respectively with positive $x$ -axis. If $27 1 \int 3 ((f^{'} (t))^{2} + 1) f^{''} (t) d t = α + β 3$ …2024 · 30 Jan · Shift 2 · Q33 · MCQ
Let $a$ and $b$ be real constants such that the function $f$ defined by $f (x) = {x^{2} + 3 x + a b x + 2, x \leq 1, x > 1$ be differentiable on $R$ . Then, the value of $\int_{- 2}^{2} f (x) d x$ …2024 · 30 Jan · Shift 2 · Q39 · MCQ
Let $f : R \to R$ be defined as $f (x) = a e^{2 x} + b e^{x} + c x$ . If $f (0) = - 1, f^{'} (lo g_{e} 2) = 21$ and $\int_{0}^{l o g_{e} 4} (f (x) - c x) d x = \frac{39}{2}$ , then the value of $∣ a + b + c ∣$ equals2024 · 30 Jan · Shift 2 · Q40 · MCQ
If the integral $525 0 \int \frac{π}{2} sin 2 x cos^{\frac{11}{2}} x (1 + cos^{\frac{5}{2}} x)^{\frac{1}{2}} d x$ is equal to $(n 2 - 64)$ , then $n$ is equal to .2024 · 31 Jan · Shift 1 · Q54 · Numerical
Let $S = (- 1, \infty)$ and $f : S \to R$ be defined as $f (x) = \int_{- 1}^{x} (e^{t} - 1)^{11} (2 t - 1)^{5} (t - 2)^{7} (t - 3)^{12} (2 t - 10)^{61} d t,$ Let $p =$ Sum of squares of the values of $x$ , where $f (x)$ …2024 · 31 Jan · Shift 1 · Q55 · Numerical
Let $f : R \to R$ be a function defined by $f (x) = \frac{4 ^{x}}{4 ^{x} + 2}$ and $M = \int_{f (a)}^{f (1 - a)} x sin^{4} (x (1 - x)) d x, N = \int_{f (a)}^{f (1 - a)} sin^{4} (x (1 - x)) d x; a e q \frac{1}{2}$ . If $α M = β N, α, β \in N$ …2024 · 31 Jan · Shift 1 · Q60 · Numerical
Let $f, g : (0, \infty) \to R$ be two functions defined by $f (x) = - x \int x (∣ t ∣ - t^{2}) e^{- t^{2}} d t$ and $g (x) = 0 \int x^{2} t^{1/2} e^{- t} d t$ . Then, the value of $9 (f (lo g_{e} 9) + g (lo g_{e} 9))$ …2024 · 31 Jan · Shift 2 · Q40 · MCQ
$\frac{120}{π ^{3}} 0 \int π \frac{x ^{2} s i n x c o s x}{s i n ^{4} x + c o s ^{4} x} d x$ is equal to .2024 · 31 Jan · Shift 2 · Q53 · Numerical
If $\int_{0}^{1} (x^{21} + x^{14} + x^{7}) (2 x^{14} + 3 x^{7} + 6)^{1/7} d x = \frac{1}{l} (11)^{m / n}$ where $l, m, n \in N, m$ and $n$ are coprime then $l + m + n$ is equal to .2023 · 1 Feb · Shift 1 · Q35 · Numerical
Let $f : R \to R$ be a differentiable function such that $f^{'} (x) + f (x) = \int_{0}^{2} f (t) d t$ . If $f (0) = e^{- 2}$ , then $2 f (0) - f (2)$ is equal to .2023 · 1 Feb · Shift 1 · Q37 · Numerical
The value of the integral $- \frac{π}{4} \int \frac{π}{4} \frac{x + \frac{π}{4}}{2 - c o s 2 x} d x$ is :2023 · 1 Feb · Shift 2 · Q30 · MCQ
If $0 \int π \frac{5 ^{c o s x} ( 1 + c o s x c o s 3 x + c o s ^{2} x + c o s ^{3} x c o s 3 x ) d x}{1 + 5 ^{c o s x}} = \frac{k π}{16}$ , then k is equal to .2023 · 1 Feb · Shift 2 · Q41 · Numerical
Let $5 f (x) + 4 f (\frac{1}{x}) = \frac{1}{x} + 3, x > 0$ . Then $18 \int_{1}^{2} f (x) d x$ is equal to :2023 · 6 Apr · Shift 1 · Q29 · MCQ
Let $f (x)$ be a function satisfying $f (x) + f (π - x) = π^{2}, \forall x \in R$ . Then $\int_{0}^{π} f (x) sin x d x$ is equal to :2023 · 6 Apr · Shift 2 · Q31 · MCQ
$lim_{n \to \infty} {(2^{\frac{1}{2}} - 2^{\frac{1}{3}}) (2^{\frac{1}{2}} - 2^{\frac{1}{5}}) \dots .. (2^{\frac{1}{2}} - 2^{\frac{1}{2 n + 1}})}$ is equal to :2023 · 6 Apr · Shift 2 · Q32 · MCQ
Let $[t]$ denote the greatest integer $\leq t$ . Then $\frac{2}{π} \int_{π /6}^{5 π /6} (8 [cosec x] - 5 [cot x]) d x$ is equal to .2023 · 8 Apr · Shift 1 · Q39 · Numerical
Let $[t]$ denote the greatest integer function. If $\int_{0}^{2.4} [x^{2}] d x = α + β 2 + γ 3 + δ 5$ , then $α + β + γ + δ$ is equal to .2023 · 8 Apr · Shift 2 · Q35 · Numerical
Let $f$ be a continuous function satisfying $\int_{0}^{t^{2}} (f (x) + x^{2}) d x = \frac{4}{3} t^{3}, \forall t > 0$ . Then $f (\frac{π ^{2}}{4})$ is equal to :2023 · 10 Apr · Shift 2 · Q26 · MCQ
The value of the integral $\int_{- l o g_{e} 2}^{l o g_{e} 2} e^{x} (lo g_{e} (e^{x} + 1 + e^{2 x})) d x$ is equal to :2023 · 11 Apr · Shift 1 · Q30 · MCQ
For $m, n > 0$ , let $α (m, n) = \int_{0}^{2} t^{m} (1 + 3 t)^{n} d t$ . If $11 α (10, 6) + 18 α (11, 5) = p (14)^{6}$ , then $p$ is equal to .2023 · 11 Apr · Shift 1 · Q43 · Numerical
If $f : R \to R$ be a continuous function satisfying $\int_{0}^{\frac{π}{2}} f (sin 2 x) sin x d x + α \int_{0}^{\frac{π}{4}} f (cos 2 x) cos x d x = 0$ , then the value of $α$ is :2023 · 11 Apr · Shift 2 · Q32 · MCQ
Let the function $f : [0, 2] \to R$ be defined as $f (x) = {e^{m i n {x^{2}, x - [x]},} e^{[x - l o g_{e} x]}, x \in [0, 1) x \in [1, 2]$ where $[t]$ denotes the greatest…2023 · 11 Apr · Shift 2 · Q35 · MCQ
If $\int_{- 0.15}^{0.15} 100 x^{2} - 1 d x = \frac{k}{3000}$ , then $k$ is equal to .2023 · 12 Apr · Shift 1 · Q39 · Numerical
$\int_{0}^{\infty} \frac{6}{e ^{3 x} + 6 e ^{2 x} + 11 e ^{x} + 6} d x =$ 2023 · 13 Apr · Shift 1 · Q30 · MCQ
Let for $x \in R, S_{0} (x) = x, S_{k} (x) = C_{k} x + k \int_{0}^{x} S_{k - 1} (t) d t$ , where $C_{0} = 1, C_{k} = 1 - \int_{0}^{1} S_{k - 1} (x) d x, k = 1, 2, 3, \dots$ Then $S_{2} (3) + 6 C_{3}$ is equal to .2023 · 13 Apr · Shift 1 · Q42 · Numerical
The value of $\frac{e ^{- \frac{π}{4}} + 0 \int \frac{π}{4} e ^{- x} t a n ^{50} x d x}{0 \int \frac{π}{4} e ^{- x} ( t a n ^{49} x + t a n ^{51} x ) d x}$ is2023 · 13 Apr · Shift 2 · Q32 · MCQ
Let $f_{n} = \int_{0}^{\frac{π}{2}} (\sum_{k = 1}^{n} sin^{k - 1} x) (\sum_{k = 1}^{n} (2 k - 1) sin^{k - 1} x) cos x d x, n \in N$ . Then $f_{21} - f_{20}$ is equal to 2023 · 13 Apr · Shift 2 · Q37 · Numerical
If $0 \int 1 \frac{1}{( 5 + 2 x - 2 x ^{2} ) ( 1 + e ^{(2 - 4 x)} )} d x = \frac{1}{α} lo g_{e} (\frac{α + 1}{β}), α, β > 0$ , then $α^{4} - β^{4}$ is equal to :2023 · 15 Apr · Shift 1 · Q25 · MCQ
The value of $12 0 \int 3 x^{2} - 3 x + 2 d x$ is 2023 · 24 Jan · Shift 1 · Q42 · Numerical
The value of $\frac{8}{π} 0 \int \frac{π}{2} \frac{( c o s x ) ^{2023}}{( s i n x ) ^{2023} + ( c o s x ) ^{2023}} d x$ is 2023 · 24 Jan · Shift 1 · Q43 · Numerical
$\frac{3 2}{4} \int \frac{3 3}{4} \frac{48}{9 - 4 x ^{2}} d x$ is equal to :2023 · 24 Jan · Shift 2 · Q36 · MCQ
Let $f$ be $a$ differentiable function defined on $[0, \frac{π}{2}]$ such that $f (x) > 0$ and $f (x) + \int_{0}^{x} f (t) 1 - (lo g_{e} f (t))^{2} d t = e, \forall x \in [0, \frac{π}{2}]$ . Then $(6 lo g_{e} f (\frac{π}{6}))^{2}$ …2023 · 24 Jan · Shift 2 · Q42 · Numerical
The minimum value of the function $f (x) = 0 \int 2 e^{∣ x - t ∣} d t$ is :2023 · 25 Jan · Shift 1 · Q25 · MCQ
The integral $16 1 \int 2 \frac{d x}{x ^{3} ( x ^{2} + 2 ) ^{2}}$ is equal to2023 · 25 Jan · Shift 2 · Q32 · MCQ
If $\frac{1}{3} \int 3 ∣ lo g_{e} x ∣ d x = \frac{m}{n} lo g_{e} (\frac{n ^{2}}{e})$ , where m and n are coprime natural numbers, then $m^{2} + n^{2} - 5$ is equal to .2023 · 25 Jan · Shift 2 · Q42 · Numerical
Let $f (x) = x + \frac{a}{π ^{2} - 4} sin x + \frac{b}{π ^{2} - 4} cos x, x \in R$ be a function which satisfies $f (x) = x + 0 \int π /2 sin (x + y) f (y) d y$ . then $(a + b)$ is equal to2023 · 29 Jan · Shift 1 · Q28 · MCQ
The value of the integral $\int_{1}^{2} (\frac{t ^{4} + 1}{t ^{6} + 1}) d t$ is2023 · 29 Jan · Shift 2 · Q34 · MCQ
The value of the integral $1/2 \int 2 \frac{t a n ^{- 1} x}{x} d x$ is equal to :2023 · 29 Jan · Shift 2 · Q36 · MCQ
If [t] denotes the greatest integer $\leq t$ , then the value of $\frac{3 ( e - 1 )}{e} 1 \int 2 x^{2} e^{[x] + [x^{3}]} d x$ is :2023 · 30 Jan · Shift 1 · Q26 · MCQ
$lim_{x \to 0} \frac{48}{x ^{4}} \int_{0}^{x} \frac{t ^{3}}{t ^{6} + 1} d t$ is equal to .2023 · 30 Jan · Shift 1 · Q33 · Numerical
Let $α \in (0, 1)$ and $β = lo g_{e} (1 - α)$ . Let $P_{n} (x) = x + \frac{x ^{2}}{2} + \frac{x ^{3}}{3} + ... + \frac{x ^{n}}{n}, x \in (0, 1)$ . Then the integral $0 \int α \frac{t ^{50}}{1 - t} d t$ …2023 · 31 Jan · Shift 1 · Q33 · MCQ
The value of $\int_{\frac{π}{3}}^{\frac{π}{2}} \frac{( 2 + 3 s i n x )}{s i n x ( 1 + c o s x )} d x$ is equal to :2023 · 31 Jan · Shift 1 · Q38 · MCQ
Let $α > 0$ . If $0 \int α \frac{x}{x + α - x} d x = \frac{16 + 20 2}{15}$ , then $α$ is equal to :2023 · 31 Jan · Shift 2 · Q27 · MCQ
If $ϕ (x) = \frac{1}{x} \frac{π}{4} \int x (42 sin t - 3 ϕ^{'} (t)) d t, x > 0$ , then $\emptyset^{'} (\frac{π}{4})$ is equal to :2023 · 31 Jan · Shift 2 · Q33 · MCQ
Let $f (θ) = sin θ + - π /2 \int π /2 (sin θ + t cos θ) f (t) d t$ . Then the value of $\int_{0}^{π /2} f (θ) d θ$ is .2022 · 24 Jun · Shift 1 · Q40 · Numerical
Let $0 \leq x \leq 2 M a x {\frac{9 - x ^{2}}{5 - x}} = α$ and $0 \leq x \leq 2 M in {\frac{9 - x ^{2}}{5 - x}} = β$ . If $β - \frac{8}{3} \int 2 α - 1 M a x {\frac{9 - x ^{2}}{5 - x}, x} d x = α_{1} + α_{2} lo g_{e} (\frac{8}{15})$ …2022 · 24 Jun · Shift 1 · Q41 · Numerical
The value of the integral $- π /2 \int π /2 \frac{d x}{( 1 + e ^{x} ) ( s i n ^{6} x + c o s ^{6} x )}$ is equal to2022 · 24 Jun · Shift 2 · Q27 · MCQ
For any real number $x$ , let $[x]$ denote the largest integer less than equal to $x$ . Let $f$ be a real valued function defined on the interval $[- 10, 10]$ by $f (x) = {x - [x], if [x] is odd 1 + [x] - x, if [x] is even .$ …2022 · 25 Jul · Shift 1 · Q29 · MCQ
Let $[t]$ denote the greatest integer less than or equal to $t$ . Then the value of the integral $\int_{- 3}^{101} ([sin (π x)] + e^{[c o s (2 π x)]}) d x$ is equal to2022 · 25 Jul · Shift 2 · Q29 · MCQ
Let $f$ be a twice differentiable function on $R$ . If $f^{'} (0) = 4$ and $f (x) + 0 \int x (x - t) f^{'} (t) d t = (e^{2 x} + e^{- 2 x}) cos 2 x + \frac{2}{a} x$ , then $(2 a + 1)^{5} a^{2}$ is equal to …2022 · 25 Jul · Shift 2 · Q42 · Numerical
Let $a_{n} = - 1 \int n (1 + \frac{x}{2} + \frac{x ^{2}}{3} + ..... + \frac{x ^{n - 1}}{n}) d x$ for every n $\in$ N. Then the sum of all the elements of the set {n} $\in$ N : an $\in$ (2, 30)} is …2022 · 25 Jul · Shift 2 · Q43 · Numerical
The value of $0 \int π \frac{e ^{c o s x} s i n x}{( 1 + c o s ^{2} x ) ( e ^{c o s x} + e ^{- c o s x} )} d x$ is equal to:2022 · 25 Jun · Shift 1 · Q24 · MCQ
If $b_{n} = \int_{0}^{\frac{π}{2}} \frac{c o s ^{2} n x}{s i n x} d x, n \in N$ , then2022 · 25 Jun · Shift 2 · Q31 · MCQ
The value of b > 3 for which $12 3 \int b \frac{1}{( x ^{2} - 1 ) ( x ^{2} - 4 )} d x = lo g_{e} (\frac{49}{40})$ , is equal to .2022 · 25 Jun · Shift 2 · Q39 · Numerical
If $n (2 n + 1) \int_{0}^{1} (1 - x^{n})^{2 n} d x = 1177 \int_{0}^{1} (1 - x^{n})^{2 n + 1} d x$ , then $n \in N$ is equal to …2022 · 26 Jul · Shift 1 · Q42 · Numerical
$0 \int 20 π (∣ sin x ∣ + ∣ cos x ∣)^{2} d x is equal to$ 2022 · 26 Jul · Shift 2 · Q27 · MCQ
Let f(x) = max {|x + 1|, |x + 2|, ....., |x + 5|}. Then $- 6 \int 0 f (x) d x$ is equal to .2022 · 26 Jun · Shift 1 · Q35 · Numerical
The value of the integral $\frac{48}{π ^{4}} 0 \int π (\frac{3 π x ^{2}}{2} - x^{3}) \frac{s i n x}{1 + c o s ^{2} x} d x$ is equal to .2022 · 26 Jun · Shift 1 · Q39 · Numerical
The integral $\frac{24}{π} \int_{0}^{2} \frac{( 2 - x ^{2} ) d x}{( 2 + x ^{2} ) 4 + x ^{4}}$ is equal to .2022 · 26 Jun · Shift 2 · Q45 · Numerical
Let $f : R \to R$ be a function defined as $f (x) = a sin (\frac{π [ x ]}{2}) + [2 - x], a \in R$ where $[t]$ is the greatest integer less than or equal to $t$ . If $x \to - 1 lim f (x)$ …2022 · 27 Jul · Shift 1 · Q30 · MCQ
Let $I = \int_{π /4}^{π /3} (\frac{8 s i n x - s i n 2 x}{x}) d x$ . Then2022 · 27 Jul · Shift 1 · Q31 · MCQ
Let a function $f : R \to R$ be defined as : $f (x) = ⎩ ⎨ ⎧ 0 \int x (5 - ∣ t - 3∣) d t, x^{2} + b x x > 4, x \leq 4$ where $b \in R$ . If $f$ is continuous…2022 · 27 Jul · Shift 1 · Q38 · MCQ
Let $f (x) = 2 + ∣ x ∣ - ∣ x - 1∣ + ∣ x + 1∣, x \in R$ . Consider $(S 1) : f^{'} (- \frac{3}{2}) + f^{'} (- \frac{1}{2}) + f^{'} (\frac{1}{2}) + f^{'} (\frac{3}{2}) = 2 (S 2) : - 2 \int 2 f (x) d x = 12$ …2022 · 27 Jul · Shift 2 · Q26 · MCQ
$0 \int 2 (2 x^{2} - 3 x + [x - \frac{1}{2}]) d x$ , where [t] is the greatest integer function, is equal to :2022 · 27 Jul · Shift 2 · Q29 · MCQ
Let $f (x) = min {[x - 1], [x - 2], \dots, [x - 10]}$ where [t] denotes the greatest integer $\leq t$ . Then $0 \int 10 f (x) d x + 0 \int 10 (f (x))^{2} d x + 0 \int 10 ∣ f (x) ∣ d x$ …2022 · 27 Jul · Shift 2 · Q38 · Numerical
Let f be a differentiable function satisfying $f (x) = \frac{2}{3} 0 \int 3 f (\frac{λ ^{2} x}{3}) d λ, x > 0$ and $f (1) = 3$ . If $y = f (x)$ passes through the point $(α, 6)$ ,…2022 · 27 Jul · Shift 2 · Q39 · Numerical
The value of the integral $- 2 \int 2 \frac{∣ x ^{3} + x ∣}{( e ^{x ∣ x ∣} + 1 )} d x$ is equal to :2022 · 27 Jun · Shift 1 · Q27 · MCQ
If m and n respectively are the number of local maximum and local minimum points of the function $f (x) = 0 \int x^{2} \frac{t ^{2} - 5 t + 4}{2 + e ^{t}} d t$ , then the ordered pair (m, n) is equal to2022 · 27 Jun · Shift 2 · Q25 · MCQ
Let f be a differentiable function in $(0, \frac{π}{2})$ . If $c o s x \int 1 t^{2} f (t) d t = sin^{3} x + cos x$ , then $\frac{1}{3} f^{'} (\frac{1}{3})$ is equal to2022 · 27 Jun · Shift 2 · Q26 · MCQ
The integral $0 \int 1 \frac{1}{7 ^{[\frac{1}{x}]}} d x$ , where [ . ] denotes the greatest integer function, is equal to2022 · 27 Jun · Shift 2 · Q27 · MCQ
The minimum value of the twice differentiable function $f (x) = 0 \int x e^{x - t} f^{'} (t) dt - (x^{2} - x + 1) e^{x}$ , $x \in R$ , is :2022 · 28 Jul · Shift 1 · Q37 · MCQ
If $0 \int 3 \frac{15 x ^{3}}{1 + x ^{2} + ( 1 + x ^{2} ) ^{3}} d x = α 2 + β 3$ , where $α, β$ are integers, then $α + β$ is equal to .2022 · 28 Jul · Shift 1 · Q41 · Numerical
Let $I_{n} (x) = \int_{0}^{x} \frac{1}{( t ^{2} + 5 ) ^{n}} d t, n = 1, 2, 3, \dots .$ Then :2022 · 28 Jul · Shift 2 · Q29 · MCQ
The value of the integral $0 \int \frac{π}{2} 60 \frac{s i n ( 6 x )}{s i n x} d x$ is equal to .2022 · 28 Jul · Shift 2 · Q39 · Numerical
Let [t] denote the greatest integer less than or equal to t. Then, the value of the integral $0 \int 1 [- 8 x^{2} + 6 x - 1] d x$ is equal to :2022 · 28 Jun · Shift 1 · Q28 · MCQ
Let f : R $\to$ R be a differentiable function such that $f (\frac{π}{4}) = 2, f (\frac{π}{2}) = 0$ and $f^{'} (\frac{π}{2}) = 1$ and let $g (x) = \int_{x}^{π /4} (f^{'} (t) sec t + tan t sec t f (t)) d t$ …2022 · 28 Jun · Shift 2 · Q28 · MCQ
Let f : R $\to$ R be a continuous function satisfying f(x) + f(x + k) = n, for all x $\in$ R where k > 0 and n is a positive integer. If $I_{1} = 0 \int 4 nk f (x) d x$ and $I_{2} = - k \int 3 k f (x) d x$ , then :2022 · 28 Jun · Shift 2 · Q29 · MCQ
The integral $0 \int \frac{π}{2} \frac{1}{3 + 2 s i n x + c o s x} d x$ is equal to :2022 · 29 Jul · Shift 1 · Q30 · MCQ
If $f (α) = 1 \int α \frac{l o g _{10} t}{1 + t} dt, α > 0$ , then $f (e^{3}) + f (e^{- 3})$ is equal to :2022 · 29 Jul · Shift 1 · Q35 · MCQ
If $[t]$ denotes the greatest integer $\leq t$ , then the value of $\int_{0}^{1} [2 x - 3 x^{2} - 5 x + 2 + 1] d x$ is :2022 · 29 Jul · Shift 2 · Q27 · MCQ
Let $f : R \to R$ be a function defined by : $f (x) = {max {t^{3} - 3 t} t \leq x x^{2} + 2 x - 6;; x \leq 2 25$ where [t] is the greatest integer…2022 · 29 Jun · Shift 1 · Q22 · MCQ
$\int_{0}^{5} cos (π (x - [\frac{x}{2}])) d x$ , where [t] denotes greatest integer less than or equal to t, is equal to:2022 · 29 Jun · Shift 1 · Q33 · MCQ
Let f be a real valued continuous function on [0, 1] and $f (x) = x + 0 \int 1 (x - t) f (t) d t$ . Then, which of the following points (x, y) lies on the curve y = f(x) ?2022 · 29 Jun · Shift 2 · Q27 · MCQ
If $0 \int 2 (2 x - 2 x - x^{2}) d x = 0 \int 1 (1 - 1 - y^{2} - \frac{y ^{2}}{2}) d y + 1 \int 2 (2 - \frac{y ^{2}}{2}) d y + I$ , then I equals2022 · 29 Jun · Shift 2 · Q28 · MCQ
Let $f (t) = 0 \int t e^{x^{3}} (\frac{x ^{8}}{( x ^{6} + 2 x ^{3} + 2 ) ^{2}}) d x$ . If $f (1) + f^{'} (1) = α e - \frac{1}{6}$ , then the value of 150 $α$ is equal to .2022 · 30 Jun · Shift 1 · Q37 · Numerical
Let f : R $\to$ R be a continuous function. Then $x \to \frac{π}{4} lim \frac{\frac{π}{4} 2 \int s e c ^{2} x f ( x ) d x}{x ^{2} - \frac{π ^{2}}{16}}$ is equal to :2021 · 1 Sep · Shift 2 · Q24 · MCQ
Let $J_{n, m} = 0 \int \frac{1}{2} \frac{x ^{n}}{x ^{m} - 1} d x$ , $\forall$ n > m and n, m $\in$ N. Consider a matrix $A = [a_{ij}]_{3 \times 3}$ where $a_{ij} = {j_{6 + i, 3} - j_{i + 3, 3}, 0, i \leq j i > j$ …2021 · 1 Sep · Shift 2 · Q30 · MCQ
The function f(x), that satisfies the condition $f (x) = x + 0 \int π /2 sin x . cos y f (y) d y$ , is :2021 · 1 Sep · Shift 2 · Q37 · MCQ
Let f : R $\to$ R be a continuous function such that f(x) + f(x + 1) = 2, for all x $\in$ R. If $I_{1} = 0 \int 8 f (x) d x$ and $I_{2} = - 1 \int 3 f (x) d x$ , then the value of I1 + 2I2 is equal to …2021 · 16 Mar · Shift 1 · Q38 · Numerical
If the normal to the curve y(x) = $0 \int x (2 t^{2} - 15 t + 10) d t$ at a point (a, b) is parallel to the line x + 3y = $-$ 5, a > 1, then the value of | a + 6b | is equal to .2021 · 16 Mar · Shift 1 · Q39 · Numerical
Consider the integral $I = \int_{0}^{10} \frac{[ x ] e ^{[x]}}{e ^{x - 1}} d x$ , where [x] denotes the greatest integer less than or equal to x. Then the value of I is equal to :2021 · 16 Mar · Shift 2 · Q30 · MCQ
Let P(x) = x2 + bx + c be a quadratic polynomial with real coefficients such that $\int_{0}^{1} P (x) d x$ = 1 and P(x) leaves remainder 5 when it is divided by (x $-$ 2). Then the value of 9(b + c) is equal to :2021 · 16 Mar · Shift 2 · Q32 · MCQ
Which of the following statements is correct for the function g( $α$ ) for $α \in$ R such that $g (α) = \frac{π}{6} \int \frac{π}{3} \frac{s i n ^{α} x}{c o s ^{α} x + s i n ^{α} x} d x$ 2021 · 17 Mar · Shift 1 · Q25 · MCQ
If [ . ] represents the greatest integer function, then the value of $0 \int \frac{π}{2} [[x^{2}] - cos x] d x$ is .2021 · 17 Mar · Shift 1 · Q40 · Numerical
Let f : R $\to$ R be defined as f(x) = e $-$ xsinx. If F : [0, 1] $\to$ R is a differentiable function with that F(x) = $\int_{0}^{x} f (t) d t$ , then the value of $\int_{0}^{1} (F^{'} (x) + f (x)) e^{x} d x$ lies in the interval2021 · 17 Mar · Shift 2 · Q23 · MCQ
If the integral $\int_{0}^{10} \frac{[ s i n 2 π x ]}{e ^{x - [x]}} d x = α e^{- 1} + β e^{- \frac{1}{2}} + γ$ , where $α$ , $β$ , $γ$ are integers and [x] denotes the greatest integer less than or equal…2021 · 17 Mar · Shift 2 · Q28 · MCQ
Let $I_{n} = \int_{1}^{e} x^{19} (lo g ∣ x ∣)^{n} d x$ , where n $\in$ N. If (20)I10 = $α$ I9 + $β$ I8, for natural numbers $α$ and $β$ , then $α - β$ equals to .2021 · 17 Mar · Shift 2 · Q38 · Numerical
Let f(x) and g(x) be two functions satisfying f(x2) + g(4 $-$ x) = 4x3 and g(4 $-$ x) + g(x) = 0, then the value of $- 4 \int 4 f (x)^{2} d x$ is2021 · 18 Mar · Shift 1 · Q41 · Numerical
Let g(x) = $\int_{0}^{x} f (t) d t$ , where f is continuous function in [ 0, 3 ] such that $\frac{1}{3} \leq$ f(t) $\leq$ 1 for all t $\in$ [0, 1] and 0 $\leq$ f(t) $\leq \frac{1}{2}$ for all t $\in$ (1, 3]. The largest possible interval in which…2021 · 18 Mar · Shift 2 · Q30 · MCQ
Let P(x) be a real polynomial of degree 3 which vanishes at x = $-$ 3. Let P(x) have local minima at x = 1, local maxima at x = $-$ 1 and $- 1 \int 1 P (x) d x$ = 18, then the sum of all the coefficients of the polynomial P(x) is…2021 · 18 Mar · Shift 2 · Q43 · Numerical
Let a be a positive real number such that $\int_{0}^{a} e^{x - [x]} d x = 10 e - 9$ where [ x ] is the greatest integer less than or equal to x. Then a is equal to:2021 · 20 Jul · Shift 1 · Q22 · MCQ
The value of the integral $- 1 \int 1 lo g_{e} (1 - x + 1 + x) d x$ is equal to:2021 · 20 Jul · Shift 1 · Q24 · MCQ
If [x] denotes the greatest integer less than or equal to x, then the value of the integral $\int_{- π /2}^{π /2} [[x] - sin x] d x$ is equal to :2021 · 20 Jul · Shift 2 · Q27 · MCQ
If the real part of the complex number $(1 - cos θ + 2 i sin θ)^{- 1}$ is $\frac{1}{5}$ for $θ \in (0, π)$ , then the value of the integral $\int_{0}^{θ} sin x d x$ is equal to:2021 · 20 Jul · Shift 2 · Q28 · MCQ
Let $g (t) = \int_{- π /2}^{π /2} cos (\frac{π}{4} t + f (x)) d x$ , where $f (x) = lo g_{e} (x + x^{2} + 1), x \in R$ . Then which one of the following is correct?2021 · 20 Jul · Shift 2 · Q35 · MCQ
If $0 \int 100 π \frac{s i n ^{2} x}{e ^{(\frac{x}{π} - [\frac{x}{π}])}} d x = \frac{α π ^{3}}{1 + 4 π ^{2}}, α \in R$ where [x] is the greatest integer less than or…2021 · 22 Jul · Shift 2 · Q28 · MCQ
$x \to 0 lim \frac{0 \int x ^{2} ( s i n t ) d t}{x ^{3}}$ is equal to :2021 · 24 Feb · Shift 1 · Q28 · MCQ
If $- a \int a (∣ x ∣ + ∣ x - 2 ∣) d x = 22$ , (a > 2) and [x] denotes the greatest integer $\leq$ x, then $- a \int a (x + [x]) d x$ is equal to…2021 · 24 Feb · Shift 1 · Q42 · Numerical
The value of the integral, $1 \int 3 [x^{2} - 2 x - 2] d x$ , where [x] denotes the greatest integer less than or equal to x, is :2021 · 24 Feb · Shift 2 · Q24 · MCQ
Let f(x) be a differentiable function defined on [0, 2] such that f'(x) = f'(2 $-$ x) for all x $\in$ (0, 2), f(0) = 1 and f(2) = e2. Then the value of $0 \int 2 f (x) d x$ is :2021 · 24 Feb · Shift 2 · Q27 · MCQ
Let f be a twice differentiable function defined on R such that f(0) = 1, f'(0) = 2 and f'(x) $e$ 0 for all x $\in$ R. If $f (x) f^{'} (x) f^{'} (x) f^{''} (x)$ = 0, for all x $\in$ …2021 · 24 Feb · Shift 2 · Q28 · MCQ
The value of $- 1 \int 1 x^{2} e^{[x^{3}]} d x$ , where [ t ] denotes the greatest integer $\leq$ t, is :2021 · 25 Feb · Shift 1 · Q26 · MCQ
If $I_{n} = \frac{π}{4} \int \frac{π}{2} cot^{n} x d x$ , then :2021 · 25 Feb · Shift 2 · Q35 · MCQ
The value of $- 2 \int 2 ∣3 x^{2} - 3 x - 6∣ d x$ is .2021 · 25 Feb · Shift 2 · Q43 · Numerical
The value of the definite integral $π /24 \int 5 π /24 \frac{d x}{1 + 3 t a n 2 x}$ is :2021 · 25 Jul · Shift 1 · Q28 · MCQ
Let $f : [0, \infty) \to [0, \infty)$ be defined as $f (x) = \int_{0}^{x} [y] d y$ where [x] is the greatest integer less than or equal to x. Which of the following is true?2021 · 25 Jul · Shift 1 · Q33 · MCQ
If $f (x) = ⎩ ⎨ ⎧ 0 \int x (5 + ∣ 1 - t ∣) d t, 5 x + 1, x > 2 x \leq 2$ , then2021 · 25 Jul · Shift 2 · Q26 · MCQ
The value of the integral $- 1 \int 1 lo g (x + x^{2} + 1) d x$ is :2021 · 25 Jul · Shift 2 · Q30 · MCQ
The value of $\frac{- 1}{2} \int \frac{1}{2} ((\frac{x + 1}{x - 1})^{2} + (\frac{x - 1}{x + 1})^{2} - 2)^{\frac{1}{2}} d x$ is :2021 · 26 Aug · Shift 1 · Q31 · MCQ
If the value of the integral $0 \int 5 \frac{x + [ x ]}{e ^{x - [x]}} d x = α e^{- 1} + β$ , where $α$ , $β \in$ R, 5 $α$ + 6 $β$ = 0, and [x] denotes the greatest integer less than or equal to x;…2021 · 26 Aug · Shift 2 · Q26 · MCQ
The value of $- \frac{π}{2} \int \frac{π}{2} (\frac{1 + s i n ^{2} x}{1 + π ^{s i n x}}) d x$ is2021 · 26 Aug · Shift 2 · Q36 · MCQ
The value of $- π /2 \int π /2 \frac{c o s ^{2} x}{1 + 3 ^{x}} d x$ is :2021 · 26 Feb · Shift 1 · Q26 · MCQ
The value of $n = 1 \sum 100 n - 1 \int n e^{x - [x]} d x$ , where [ x ] is the greatest integer $\leq$ x, is :2021 · 26 Feb · Shift 1 · Q29 · MCQ
The value of the integral $0 \int π ∣ sin 2 x ∣ d x$ is .2021 · 26 Feb · Shift 1 · Q37 · Numerical
Let $f (x) = 0 \int x e^{t} f (t) d t + e^{x}$ be a differentiable function for all x $\in$ R. Then f(x) equals :2021 · 26 Feb · Shift 2 · Q31 · MCQ
For x > 0, if $f (x) = 1 \int x \frac{l o g _{e} t}{( 1 + t )} d t$ , then $f (e) + f (\frac{1}{e})$ is equal to :2021 · 26 Feb · Shift 2 · Q32 · MCQ
If $I_{m, n} = 0 \int 1 x^{m - 1} (1 - x)^{n - 1} d x$ , for m, $n \geq 1$ , and $0 \int 1 \frac{x ^{m - 1} + x ^{n - 1}}{( 1 + x ) ^{m + 1}} d x = α I_{m, n} α \in R$ , then $α$ equals …2021 · 26 Feb · Shift 2 · Q45 · Numerical
$6 \int 16 \frac{l o g _{e} x ^{2}}{l o g _{e} x ^{2} + l o g _{e} ( x ^{2} - 44 x + 484 )} d x$ is equal to :2021 · 27 Aug · Shift 1 · Q32 · MCQ
The value of the integral $0 \int 1 \frac{x d x}{( 1 + x ) ( 1 + 3 x ) ( 3 + x )}$ is :2021 · 27 Aug · Shift 2 · Q34 · MCQ
The value of the definite integral $- \frac{π}{4} \int \frac{π}{4} \frac{d x}{( 1 + e ^{x c o s x} ) ( s i n ^{4} x + c o s ^{4} x )}$ is equal to :2021 · 27 Jul · Shift 1 · Q25 · MCQ
Let the domain of the function $f (x) = lo g_{4} (lo g_{5} (lo g_{3} (18 x - x^{2} - 77)))$ be (a, b). Then the value of the integral $a \int b \frac{s i n ^{3} x}{( s i n ^{3} x + s i n ^{3} ( a + b - x )} d x$ …2021 · 27 Jul · Shift 1 · Q42 · Numerical
Let $F : [3, 5] \to R$ be a twice differentiable function on (3, 5) such that $F (x) = e^{- x} 3 \int x (3 t^{2} + 2 t + 4 F^{'} (t)) d t$ . If $F^{'} (4) = \frac{α e ^{β} - 224}{( e ^{β} - 4 ) ^{2}}$ , then $α$ + $β$ is…2021 · 27 Jul · Shift 1 · Q44 · Numerical
Let f : (a, b) $\to$ R be twice differentiable function such that $f (x) = \int_{a}^{x} g (t) d t$ for a differentiable function g(x). If f(x) = 0 has exactly five distinct roots in (a, b), then g(x)g'(x) = 0 has at least :2021 · 27 Jul · Shift 2 · Q37 · MCQ
If $\int_{0}^{π} (sin^{3} x) e^{- s i n^{2} x} d x = α - \frac{β}{e} \int_{0}^{1} t e^{t} d t$ , then $α$ + $β$ is equal to .2021 · 27 Jul · Shift 2 · Q39 · Numerical
Let f be a non-negative function in [0, 1] and twice differentiable in (0, 1). If $\int_{0}^{x} 1 - (f^{'} (t))^{2} d t = \int_{0}^{x} f (t) d t$ , $0 \leq x \leq 1$ and f(0) = 0, then $x \to 0 lim \frac{1}{x ^{2}} \int_{0}^{x} f (t) d t$ …2021 · 31 Aug · Shift 1 · Q23 · MCQ
Let [t] denote the greatest integer $\leq$ t. Then the value of $8. - \frac{1}{2} \int 1 ([2 x] + ∣ x ∣) d x$ is .2021 · 31 Aug · Shift 1 · Q37 · Numerical
If $x ϕ (x) = 5 \int x (3 t^{2} - 2 ϕ^{'} (t)) d t$ , x > $-$ 2, and $ϕ$ (0) = 4, then $ϕ$ (2) is .2021 · 31 Aug · Shift 1 · Q43 · Numerical
If [x] is the greatest integer $\leq$ x, then $π^{2} 0 \int 2 (sin \frac{π x}{2}) (x - [x])^{[x]} d x$ is equal to :2021 · 31 Aug · Shift 2 · Q35 · MCQ
The integral $0 \int 2 ∣ ∣ x - 1 ∣ - x ∣ d x$ is equal to .2020 · 2 Sep · Shift 1 · Q30 · Numerical
Let [t] denote the greatest integer less than or equal to t. Then the value of $1 \int 2 ∣ 2 x - [3 x] ∣ d x$ is .2020 · 2 Sep · Shift 2 · Q32 · Numerical
$- π \int π ∣ π - ∣ x ∣ ∣ d x$ is equal to :2020 · 3 Sep · Shift 1 · Q29 · MCQ
Suppose f(x) is a polynomial of degree four, having critical points at –1, 0, 1. If T = {x $\in$ R | f(x) = f(0)}, then the sum of squares of all the elements of T is :2020 · 3 Sep · Shift 2 · Q22 · MCQ