Definite Integrals — JEE Main PYQs

229 previous year questions on Definite Integrals (Mathematics), newest first. Open any question for its complete step-by-step solution.

The value of $\int_{- π /2}^{π /2} \frac{c o s ^{2} x}{1 + 3 ^{x}} d x$ is :
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The value of $\sum_{n = 1}^{100} \int_{n - 1}^{n} e^{x - [x]} d x$ , where $[x]$ is the greatest integer $\leq x$ , is:
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The value of the integral $\int_{0}^{π} ∣ sin 2 x ∣ d x$ is ______.
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$\int_{6}^{16} \frac{l o g _{e} x ^{2}}{l o g _{e} x ^{2} + l o g _{e} ( x ^{2} - 44 x + 484 )} d x$ is equal to
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For $x > 0$ , if $f (x) = \int_{1}^{x} \frac{l o g _{e} t}{( 1 + t )} d t$ , then $f (e) + f (\frac{1}{e})$ is equal to :
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If $I_{m, n} = \int_{0}^{1} x^{m - 1} (1 - x)^{n - 1} d x$ , for $m, n \geq 1$ , and $\int_{0}^{1} \frac{x ^{m - 1} + x ^{n - 1}}{( 1 + x ) ^{m + n}} d x = α I_{m, n}, α \in R$ , then $α$ equals.....
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If $[x]$ is the greatest integer $\leq x$ , then $π^{2} \int_{0}^{2} (sin \frac{π x}{2}) (x - [x])^{[x]} d x$ is equal to:
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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
$x \to 0 lim \frac{1}{x ^{2}} \int_{0}^{x} f (t) d t$
:
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Let
$[t]$
denote the greatest integer
$\leq t$
. Then the value of
$8 \cdot \int_{- \frac{1}{2}}^{1} ([2 x] + ∣ x ∣) d x$
is ______.
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If $x ϕ (x) = \int_{5}^{x} (3 t^{2} - 2 ϕ^{'} (t)) d t, x > - 2$ , and $ϕ (0) = 4$ , then $ϕ (2)$ is ____.
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The value of the definite integral $\int_{- \frac{π}{4}}^{\frac{π}{4}} \frac{d x}{( 1 + e ^{x c o s x} ) ( s i n ^{4} x + c o s ^{4} x )}$ is equal to :
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Let $F : [3, 5] \to R$ be a twice differentiable function on ( 3,5 ) such that $F (x) = e^{- x} \int_{3}^{x} (3 t^{2} + 2 t + 4 F^{'} (t)) d t$ .
If $F^{'} (4) = \frac{α e ^{β} - 224}{( e ^{β} - 4 ) ^{2}}$ , then $α + β$ …
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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 $\int_{a}^{b} \frac{s i n ^{3} x}{( s i n ^{3} x + s i n ^{3} ( a + b - x ) )} d x$ is equal to ____.
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Which of the following statements is incorrect for the function $g (α)$ for $α \in R$ such that $g (α) = \int_{\frac{π}{6}}^{\frac{π}{3}} \frac{s i n ^{α} x}{c o s ^{α} x + s i n ^{α} x} d x$
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If [.] represents the greatest integer function, then the value of $\int_{0}^{\frac{π}{2}} [[x^{2}] - cos x] d x$ is ____
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The value of the definite integral
$\int_{π /24}^{5 π /24} \frac{d x}{1 + 3 t a n 2 x}$ is :
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The value of the integral $\int_{- 1}^{1} lo g (x + x^{2} + 1) d x$ is :
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If $\int_{0}^{π} (sin^{3} x) e^{- s i n^{2} x} dx = α - \frac{β}{e} \int_{0}^{1} t e^{t} dt$ , then $α + β$ is equal to ______
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The function $f (x)$ , that satisfies the condition $f (x) = x + \int_{0}^{π /2} sin x \cdot cos y f (y) d y$ is:
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Let $f : R \to R$ be a continuous function such that $f (x) + f (x + 1) = 2$ , for all $x \in R$ . If $I_{1} = \int_{0}^{8} f (x) d x$ and $I_{2} = \int_{- 1}^{3} f (x) d x$ , then the value of $I_{1} + 2 I_{2}$ is equal to ____
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Let $f : (0, 2) \to R$ be defined as $f (x) = lo g_{2} (1 + tan (\frac{π x}{4}))$ . Then, $lim_{n \to \infty} \frac{2}{n} (f (\frac{1}{n}) + f (\frac{2}{n}) + \dots . + f (1))$ is equal to
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Consider the integral
$I = \int_{0}^{10} \frac{[ x ] e ^{[x]}}{e ^{x - 1}} dx$ , where $[x]$ denotes the greatest integer less than or equal to x. Then the value of I is equal to :
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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 $g (3)$ lies is :
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Let $f : R \to R$ be defined as $f (x) = e^{- x} sin x$ . If $F : [0, 1] \to R$ is a differentiable function such 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 interval
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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 to $x$ , then the value of $α + β + γ$ is equal …
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Let $f (x)$ and $g (x)$ be two functions satisfying $f (x^{2}) + g (4 - x) = 4 x^{3}$ and $g (4 - x) + g (x) = 0$ , the value of $\int_{- 4}^{4} f (x^{2}) d x$ is ____.
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If $f (a + b + 1 - x) = f (x)$ , for all $x$ , where $a$ and $b$ are fixed positive real numbers, then $\frac{1}{a + b} \int_{a}^{b} x (f (x) + f (x + 1)) d x$ is
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The value of $α$ for which $4 α \int_{- 1}^{2} e^{- α ∣ x ∣} d x = 5$ , is
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If $I = \int_{1}^{2} \frac{d x}{2 x ^{3} - 9 x ^{2} + 12 x + 4}$ , then:
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The value of $\int_{0}^{2 π} \frac{x s i n ^{8} x}{s i n ^{8} x + c o s ^{8} x} d x$ is equal to:
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If for all real triplets $(a, b, c)$ , $f (x) = a + b x + c x^{2}$ ; then $\int_{0}^{1} f (x) d x$ is equal to:
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$\int_{- π}^{π} ∣ π - ∣ x ∣∣ d x$ is equal to:
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The value of $\int_{- π /2}^{π /2} \frac{1}{1 + e ^{s i n x}} dx$ is:
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If $I_{1} = \int_{0}^{1} (1 - x^{50})^{100} d x$ and $I_{2} = \int_{0}^{1} (1 - x^{50})^{101} d x$ such that $I_{2} = α I_{1}$ then $α$ equals to:
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Let $f (x) = ∣ x - 2∣$ and $g (x) = f (f (x)), x \in [0, 4]$ . Then $\int_{0}^{3} (g (x) - f (x)) d x$ is equal to:
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The integral $\int_{π /6}^{π /3} tan^{3} x \cdot sin^{2} 3 x (2 sec^{2} x \cdot sin^{2} 3 x + 3 tan x \cdot sin 6 x) d x$ is
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Let ${x}$ and $[x]$ denote the fractional part of x and the greatest integer $\leq x$ respectively of a real number $x$ . if $\int_{0}^{n} {x} d x$ , $\int_{0}^{n} [x] d x$ and $10 (n^{2} - n)$ , $(n \in N, n > 1)$ are three consecutive terms of a …
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If $f (x) = \frac{2 - x c o s x}{2 + x c o s x}$ and $g (x) = lo g_{e} x, (x > 0)$ then the value of the integral $\int_{- π /4}^{π /4} g (f (x)) d x$ is:
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Let $f (x) = \int_{0}^{x} g (t) d t$ , where $g$ is a non zero even function. If $f (x + 5) = g (x)$ , then $\int_{0}^{x} f (t) d t$ equals
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The value of $\int_{0}^{π /2} \frac{s i n ^{3} x}{s i n x + c o s x} d x$ is:
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Let $I = \int_{a}^{b} (x^{4} - 2 x^{2}) dx$ . If I is minimum then the ordered pair $(a, b)$ is:
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The integral $\int_{π /6}^{π /4} \frac{d x}{s i n 2 x ( t a n ^{5} x + c o t ^{5} x )}$ equals:
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If $\int_{0}^{\frac{π}{2}} \frac{c o t x}{c o t x + c o s ec x} d x = m (π + n)$ , then $m \cdot n$ is equal to
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The integral $\int_{π /6}^{π /3} sec^{2/3} x cosec^{4/3} x d x$ is equal to
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The value of the integral $\int_{- 2}^{2} \frac{s i n ^{2} x}{[ \frac{x}{π} ] + \frac{1}{2}} d x$ (where $[x]$ denotes the greatest integer less than or equal to $x$ ) is:
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Let $f$ and $g$ be continuous functions on $[0, a]$ such that $f (x) = f (a - x)$ and $g (x) + g (a - x) = 4$ , then $\int_{0}^{a} f (x) g (x) d x$ is equal to:
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A value of $α$ such that $\int_{α}^{α + 1} \frac{d x}{( x + α ) ( x + α + 1 )} = lo g_{e} (\frac{9}{8})$ is:
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The integral $\int_{1}^{e} {(\frac{x}{e})^{2 x} - (\frac{e}{x})^{x}} lo g_{e} x d x$ is equal to :
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If $f : R \to R$ is a differentiable function and $f (2) = 6$ , then $lim_{x \to 2} \int_{6}^{f (x)} \frac{2 t d t}{( x - 2 )}$ is:
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The value of the integral $\int_{0}^{1} x cot^{- 1} (1 - x^{2} + x^{4}) d x$ is
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