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 $\int_{- π}^{π} \frac{2 y ( 1 + s i n y )}{1 + c o s ^{2} y} d y$ is
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Let $β (m, n) = \int_{0}^{1} x^{m - 1} (1 - x)^{n - 1} d x, m, n > 0$ . If $\int_{0}^{1} (1 - x^{10})^{20} d x = a \times β (b, c)$ , then $100 (a + b + c)$ equals ____ .
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If $f (t) = \int_{0}^{π} \frac{2 x d x}{1 - c o s ^{2} t s i n ^{2} x}, 0 < t < π$ , then the value of $\int_{0}^{\frac{π}{2}} \frac{π ^{2} d t}{f ( t )}$ equals ____ .
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$\int_{0}^{π /4} \frac{c o s ^{2} x s i n ^{2} x}{( c o s ^{3} x + s i n ^{3} x ) ^{2}} d x$ is equal to
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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 ____.
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Let $[t]$ denote the largest integer less than or equal to $t$ .

If $\int_{0}^{3} ([x^{2}] + [\frac{x ^{2}}{2}]) d x = a + b 2 - 3 - 5 + c 6 - 7$ where $a, b, c \in Z$ , then $a + b + c$ is equal to ______.
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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 $147 I_{20} = 148 I_{21}$ is :
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Let $\int_{α}^{l o g_{e} 4} \frac{dx}{e ^{x} - 1} = \frac{π}{6}$ . Then $e^{α}$ and $e^{- α}$ are the roots of the equation :
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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 ______.
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Let $f, g : (0, \infty) \to R$ be two functions defined by $f (x) = \int_{- x}^{x} (∣ t ∣ - t^{2}) e^{- t^{2}} d t$ and $g (x) = \int_{0}^{x^{2}} t^{1/2} e^{- t} d t$ . Then the value of …
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$\frac{120}{π ^{3}} \int_{0}^{π} \frac{x ^{2} s i n x c o s x}{s i n ^{4} x + c o s ^{4} x} dx$ is equal to ______.
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The value $9 \int_{0}^{9} [\frac{10 x}{x + 1}] d x$ , where $[t]$ denotes the greatest integer less than or equal to $t$ , is ______.
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For $0 < a < 1$ , the value of the integral $\int_{0}^{π} \frac{d x}{1 - 2 a cos x + a ^{2}}$ is :
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Let $f (x) = \int_{0}^{x} g (t) lo g_{e} (\frac{1 - t}{1 + t}) d t$ , where g is a continuous odd function.

If $\int_{- π /2}^{π /2} (f (x) + \frac{x ^{2} c o s x}{1 + e ^{x}}) d x = (\frac{π}{α})^{2} - α$ , then $α$ is equal to
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If $(a, b)$ be the orthocentre of the triangle whose vertices are $(1, 2), (2, 3)$ and $(3, 1)$ , and $I_{1} = \int_{a}^{b} x sin (4 x - x^{2}) d x$ , $I_{2} = \int_{a}^{b} sin (4 x - x^{2}) d x$ , then $36 \frac{I _{1}}{I _{2}}$ is equal to :
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If $\int_{0}^{1} \frac{1}{3 + x + 1 + x} d x = a + b 2 + c 3$ , where $a, b, c$ are rational numbers, then $2 a + 3 b - 4 c$ is equal to :
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The value of $\int_{0}^{1} (2 x^{3} - 3 x^{2} - x + 1)^{\frac{1}{3}} d x$ is equal to:
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If $\int_{0}^{\frac{π}{3}} cos^{4} x d x = aπ + b 3$ , where $a$ and $b$ are rational numbers, then $9 a + 8 b$ is equal to :
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Let $f : (0, \infty) \to R$ and $F (x) = \int_{0}^{x} t f (t) d t$ . If $F (x^{2}) = x^{4} + x^{5}$ , then $\sum_{r = 1}^{12} f (r^{2})$ is equal to :
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If the integral $525 \int_{0}^{\frac{π}{2}} sin 2 x cos^{\frac{11}{2}} x (1 + cos^{\frac{5}{2}} x)^{\frac{1}{2}} d x$ is equal to $(n 2 - 64)$ , then $n$ is equal to
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Let $f : R \to R$ be a function defined by $f (x) = \frac{4 ^{x}}{4 ^{x} + 2}$ and $M = \int_{f (a)}^{f (1 - a)} x sin^{4} (x (1 - x)) d x$ , $N = \int_{f (a)}^{f (1 - a)} sin^{4} (x (1 - x)) d x$ ; $a \neq = \frac{1}{2}$ . If $α M = β N$ , $α, β \in N$ , then the le …
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The value of the integral
$\int_{0}^{\frac{π}{4}} \frac{x d x}{s i n ^{4} ( 2 x ) + c o s ^{4} ( 2 x )}$ equals :
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If $\int_{- π /2}^{π /2} \frac{8 2 c o s x d x}{( 1 + e ^{s i n x} ) ( 1 + s i n ^{4} x )} = α π + β lo g_{e} (3 + 22)$ , where $α, β$ are integers, then $α^{2} + β^{2}$ equals ______.
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If the value of the integral
$\int_{- \frac{π}{2}}^{\frac{π}{2}} (\frac{x ^{2} c o s x}{1 + π ^{x}} + \frac{1 + s i n ^{2} x}{1 + e ^{s i n x^{2023}}}) d x = \frac{π}{4} (π + a) - 2$ ,
then the value of a is
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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 $\frac{π}{6}$ and $\frac{π}{4}$ , respectively with positive x -axis. If …
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Let $f : R \to R$ be defined $f (x) = a e^{2 x} + b e^{x} + c x$ . If $f (0) = - 1, f^{'} (lo g_{e} 2) = 21$ and $\int_{0}^{l o g_{c} 4} (f (x) - c x) d x = \frac{39}{2}$ , then the value of $∣ a + b + c ∣$ equals :
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Let $f : R \to R$ be a function defined $f (x) = \frac{x}{( 1 + x ^{4} ) ^{1/4}}$ and $g (x) = f (f (f (f (x))))$ then $18 \int_{0}^{25} x^{2} g (x) dx$
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Let $f (x)$ be a function satisfying $f (x) + f (π - x) = π^{2}, \forall x \in R$ . Then $\int_{0}^{π} f (x) sin x dx$ is equal to
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Let $f (x) = \frac{x}{( 1 + x ^{n} ) ^{1/ n}}, x \in R - {- 1}, n \in N, n > 2$ . If $f^{n} (x) =$ (fofof ...... upto n times) ( x ), then …
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Let $[t]$ denotes 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
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Let $5 f (x) + 4 f (\frac{1}{x}) = \frac{1}{x} + 3, x > 0$ . Then $18 \int_{1}^{2} f (x) dx$ is equal to:
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Let $[t]$ denote the greatest integer function. If $\int_{0}^{2.4} [x^{2}] d x = α + β 2 + γ 3 + δ 5$ , where $α, β, γ, δ$ are rational numbers, then $α + β + γ + δ$ is equal to ______.
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Let $f$ be a continuous function satisfying $\int_{0}^{t^{2}} (f (x) + x^{2}) dx = \frac{4}{3} t^{3}, \forall t > 0$ . Then $f (\frac{π ^{2}}{4})$ is equal to :
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If $\int_{- 0.15}^{0.15} 100 x^{2} - 1 dx = \frac{k}{3000}$ , then k is equal to ______.
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The value of $\frac{e ^{- \frac{π}{4}} + \int _{0}^{\frac{π}{4}} e ^{- x} t a n ^{50} x d x}{\int _{0}^{\frac{π}{4}} e ^{- x} ( t a n ^{49} x + t a n ^{51} x ) d x}$ is
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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 ______.
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$\int_{0}^{\infty} \frac{6}{e ^{3 x} + 6 e ^{2 x} + 11 e ^{x} + 6} dx$
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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 ______.
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If
$\int_{0}^{1} \frac{1}{( 5 + 2 x - 2 x ^{2} ) ( 1 + e ^{(2 - 4 x)} )} dx = \frac{1}{α} lo g_{e} (\frac{α + 1}{β})$ ,
$α, β > 0$ , then $α^{4} - β^{4}$ is equal to:
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Let the function $f : [0, 2] \to R$ be defined as …
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If $f : R \to R$ be a continuous function satisfying
$\int_{0}^{π /2} f (sin 2 x) \cdot sin x d x + α \int_{0}^{π /4} f (cos 2 x) \cdot cos x d x = 0$ , then $α$ is equal to
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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
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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
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The value of $\frac{8}{π} \int_{0}^{\frac{π}{2}} \frac{( c o s x ) ^{2023}}{( s i n x ) ^{2023} + ( c o s x ) ^{2023}} d x$ is
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If $[t]$ denotes the greatest integer $\leq t$ , then the value of $\frac{3 ( e - 1 )}{e} \int_{1}^{2} x^{2} e^{[x] + [x^{3}]} dx$ is:
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The value of the integral $\int_{- \frac{π}{4}}^{\frac{π}{4}} \frac{x + \frac{π}{4}}{2 - c o s 2 x} d x$ is :
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If $\int_{0}^{π} \frac{5 ^{c o s x} ( 1 + c o s x c o s 3 x + c o s ^{2} x + c o s ^{3} x c o s 3 x ) d x}{1 + 5 ^{c o s x}} = \frac{k π}{16}$ , then k is equal to ______.
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The value of the integral $\int_{1}^{2} (\frac{t ^{4} + 1}{t ^{6} + 1}) d t$ is :
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The value of the integral $\int_{1/2}^{2} \frac{t a n ^{- 1} x}{x} d x$ is equal to
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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 $f (x) = x + \int_{0}^{π /2} sin (x + y) f (y) d y$ . Then $(a + b)$ is equal to
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