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 $\int_{- π /6}^{π /6} (\frac{π + 4 x ^{11}}{1 - s i n ( ∣ x ∣ + π /6 )}) d x$ is equal to
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$6 \int_{0}^{π} ∣ (sin 3 x + sin 2 x + sin x) ∣ d x$ is equal to.......
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Let $[\cdot]$ denote the greatest integer function and $f (x) = lim_{n \to \infty} \frac{1}{n ^{3}} \sum_{k = 1}^{n} [\frac{k ^{2}}{3 ^{x}}]$ . Then $12 \sum_{j = 1}^{\infty} f (j)$ is equal …
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If $\int_{0}^{1} 4 cot^{- 1} (1 - 2 x + 4 x^{2}) dx = a tan^{- 1} (2) - b lo g_{e} (5)$ , where $a, b \in N$ , then $(2 a + b)$ is equal to ____.
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Let $f : [1, \infty) \to R$ be a differentiable function. If $6 \int_{1}^{x} f (t) d t = 3 x f (x) + x^{3} - 4$ for all $x \geq 1$ , then the value of $f (2) - f (3)$ is
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The value of $\int_{- \frac{π}{2}}^{\frac{π}{2}} (\frac{1}{[ x ] + 4}) dx$ , where [•] denotes the greatest integer function, is
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Let $[∙]$ be the greatest integer function. If $α = \int_{0}^{64} (x^{1/3} - [x^{1/3}]) dx$ , then …
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The value of the integral $\int_{\frac{π}{24}}^{\frac{5 π}{24}} \frac{dx}{1 + 3 t a n 2 x}$ is :
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Let f be a twice differentiable non-negative function such that
$(f (x))^{2} = 25 + \int_{0}^{x} ((f (t))^{2} + (f^{'} (t))^{2}) d t$

Then the mean of $f (lo g_{e} (1)), f (lo g_{e} (2)), \dots, f (lo g_{e} (625))$ is equal to ______.
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The number of elements in the set $S = {x : x \in [0, 100] and \int_{0}^{x} t^{2} sin (x - t) d t = x^{2}}$ is
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Let a differentiable function $f$ satisfy the equation $\int_{0}^{36} f (\frac{tx}{36}) dt = 4 α f (x)$ .

If $y = f (x)$ is a standard parabola passing through the points $(2, 1)$ and $(- 4, β)$ ,

Then …
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If $f (x)$ satisfies the relation $f (x) = e^{x} + \int_{0}^{1} (y + xe^{x}) f (y) d y$ , then $e + f (0)$ is equal to ______.
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Let $f$ be a polynomial function such that
$f (x^{2} + 1) = x^{4} + 5 x^{2} + 2$ , for all $x \in R$ .
Then $\int_{0}^{3} f (x) dx$ is equal to
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The value of $\sum_{r = 1}^{20} (π (\int_{0}^{r} x ∣ sin π x ∣ dx))$ is ______.
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Let [ • ] denote the greatest integer function. Then $\int_{- \frac{π}{2}}^{\frac{π}{2}} (\frac{12 ( 3 + [ x ])}{3 + [ s i n x ] + [ c o s x ]}) d x$ is equal to:
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Let [•] denote the greatest integer function. Then the value of $\int_{0}^{3} (\frac{e ^{x} + e ^{- x}}{[ x ]!}) d x$ is:
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If $α = \int_{0}^{23} lo g_{2} (x^{2} + 4) d x + \int_{2}^{4} 2^{x} - 4 d x$ , then $α^{2}$ is equal to ____.
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If $α = 1$ and $β = 1 + i 2$ , where $i = - 1$ are two roots of the equation
$x^{3} + ax^{2} + bx + c = 0, a, b, c \in R$ , then $\int_{- 1}^{1} (x^{3} + a x^{2} + b x + c) d x$ is e …
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The integral $\int_{0}^{1} cot^{- 1} (1 + x + x^{2}) dx$ is equal to :
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Let $f$ be a twice differentiable function such that
$f (x) = \int_{0}^{x} tan (t - x) dt - \int_{0}^{x} f (t) tan tdt, x \in (- \frac{π}{2}, \frac{π}{2})$
Then …
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Let $\int_{- 2}^{2} (∣ sin x ∣ + [x sin x]) d x = 2 (3 - cos 2) + β$ , where $[\cdot]$ is the greatest integer function. Then $β sin (\frac{β}{2})$ equals.
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The value of the integral $\int_{0}^{\infty} \frac{l o g _{e} ( x )}{x ^{2} + 4} dx$ is:
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The value of the integral $\int_{\frac{π}{6}}^{\frac{π}{3}} (\frac{4 - cosec ^{2} x}{c o s ^{4} x}) d x$ is :
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Let $f : [1, \infty) \to R$ be a differentiable function defined as $f (x) = \int_{1}^{x} f (t) d t + (1 - x) (lo g_{e} x - 1) + e$ . Then the value of $f (f (1))$ is:
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The value of the integral $\int_{- 1}^{1} (\frac{x ^{3} + ∣ x ∣ + 1}{x ^{2} + 2∣ x ∣ + 1}) d x$ is equal to
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The value of the integral $\int_{- \frac{π}{4}}^{\frac{π}{4}} (\frac{32 c o s ^{4} x}{1 + e ^{s i n x}}) d x$ is :
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The value of the integral
$\int_{0}^{2} \frac{x ( x ^{2} + x + 1 )}{( x + 1 ) ( x ^{4} + x ^{2} + 1 )} d x$ is equal to :
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If $\int_{π /6}^{π /4} (cot (x - \frac{π}{3}) cot (x + \frac{π}{3}) + 1) d x = α lo g_{e} (3 - 1)$ ,
then $9 α^{2}$ is equal to ____.
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The value of $\int_{0}^{20 π} (sin^{4} x + cos^{4} x) dx$ is equal to:
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Let for $f (x) = 7 tan^{8} x + 7 tan^{6} x - 3 tan^{4} x - 3 tan^{2} x$ , $I_{1} = \int_{0}^{π /4} f (x) dx$ and $I_{2} = \int_{0}^{π /4} x f (x) dx$ . Then …
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The value of …
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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 $\int_{0}^{π /2} \frac{x s i n x c o s x}{s i n ^{4} x + c o s ^{4} x} d x$ equals:
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If $24 \int_{0}^{\frac{π}{4}} (sin 4 x - \frac{π}{12} + [2 sin x]) d x = 2 π + α$ , where [•] denotes the greatest integer function, then $α$ is equal to ____.
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Let $f$ be a real valued continuous function defined on the positive real axis such that $g (x) = \int_{0}^{x} t f (t) dt$ . If $g (x^{3}) = x^{6} + x^{7}$ , then value of …
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Let $f : R \to R$ be a twice differentiable function such that $f (2) = 1$ . If $F (x) = x f (x)$ for all $x \in R$ , $\int_{0}^{2} x F^{'} (x) d x = 6$ and …
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In $I (m, n) = \int_{0}^{1} x^{m - 1} (1 - x)^{n - 1} dx, m, n > 0$ , then $I (9, 14) + I (10, 13)$ is
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Let $f : [1, \infty) \to [2, \infty)$ be a differentiable function. If $10 \int_{1}^{x} f (t) dt = 5 x f (x) - x^{5} - 9$ for all $x \geq 1$ , then the value of $f (3)$ is:
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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:
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$4 \int_{0}^{1} (\frac{1}{3 + x ^{2} + 1 + x ^{2}}) d x - 3 lo g_{e} (3)$ is equal to :
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The integral $\int_{0}^{π} \frac{8 x d x}{4 c o s ^{2} x + s i n ^{2} x}$ is equal to
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Let [ • ] denote the greatest integer function. If $\int_{0}^{e^{3}} [\frac{1}{e ^{x - 1}}] dx = α - lo g_{e} 2$ , then $α^{3}$ is equal to ______.
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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 $β = \int_{1}^{2} g (x) dx$ , then the value of $9 α + β$ is:
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The value of $\int_{- 1}^{1} \frac{( 1 + ∣ x ∣ - x ) e ^{x} + ( ∣ x ∣ - x ) e ^{- x}}{e ^{x} + e ^{- x}} d x$ is equal to
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The integral $\int_{0}^{π} \frac{( x + 3 ) s i n x}{1 + 3 c o s ^{2} x} d x$ is equal to :
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Let $f (x)$ be a positive function and
$I_{1} = \int_{- \frac{1}{2}}^{1} 2 x f (2 x (1 - 2 x)) d x$ and $I_{2} = \int_{- 1}^{2} f (x (1 - x)) d x$ .
Then the value of $\frac{I _{2}}{I _{1}}$ is equal to
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The integral $\int_{- 1}^{\frac{3}{2}} (π^{2} x sin (π x)) dx$ is equal to
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If ∫_{-π/2}^{π/2} (96 x² cos²x)/(1+eˣ) dx = απ³ + βπ with α, β ∈ Z, then (α+β)² equals:
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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
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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 $a + b$ is equal to ____.
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If the value of the integral $\int_{- 1}^{1} \frac{cos α x}{1 + 3 ^{x}} d x$ is $\frac{2}{π}$ . Then, a value of $α$ is
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