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.

$\int_{\frac{3 2}{4}}^{\frac{3 3}{4}} \frac{48}{9 - 4 x ^{2}} d x$ is equal to
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If $ϕ (x) = \frac{1}{x} \int_{\frac{π}{4}}^{x} (42 sin t - 3 ϕ^{'} (t)) dt, x > 0$ , then $ϕ^{'} (\frac{π}{4})$ is equal to :
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Let $α > 0$ . If $\int_{0}^{α} \frac{x}{x + α - x} dx = \frac{16 + 20 2}{15}$ , then $α$ is equal to :
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$lim_{n \to \infty} \frac{3}{n} {4 + (2 + \frac{1}{n})^{2} + (2 + \frac{2}{n})^{2} + \dots + (3 - \frac{1}{n})^{2}}$ is equal to
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Let $α \in (0, 1)$ and $β = lo g_{e} (1 - α)$ . Let $P_{n} (x) = x + \frac{x ^{2}}{2} + \frac{x ^{3}}{3} + \dots .. + \frac{x ^{n}}{n}, x \in (0, 1)$ .

Then the integral $\int_{0}^{α} \frac{t ^{50}}{1 - t} dt$ is equal to
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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
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If $[t]$ denotes the greatest integer $\leq t$ , then the value of $\int_{0}^{1} [2 x - 3 x^{2} - 5 x + 2 + 1] dx$ is:
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If $\int_{0}^{3} \frac{15 x ^{3}}{1 + x ^{2} + ( 1 + x ^{2} ) ^{3}} dx = α 2 + β 3$ , where $α, β$ are integers, then $α + β$ is equal to ____.
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The value of the integral $\int_{0}^{\frac{π}{2}} 60 \frac{s i n ( 6 x )}{s i n x} d x$ is equal to
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$I = \int_{π /4}^{π /3} (\frac{8 s i n x - s i n 2 x}{x}) d x$ . Then
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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 …
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$\int_{0}^{20 π} (∣ sin x ∣ + ∣ cos x ∣)^{2} d x$ is equal to:
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$\int_{0}^{2} (2 x^{2} - 3 x + [x - \frac{1}{2}]) d x$ ,
where $[t]$ is the greatest integer function, is equal to:
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Let $f (x) = min {[x - 1], [x - 2], \dots, [x - 10]}$ where $[t]$ denotes the greatest integer $\leq t$ . Then $\int_{0}^{10} f (x) d x + \int_{0}^{10} (f (x))^{2} d x + \int_{0}^{10} ∣ f (x) ∣ d x$ is equal to ____.
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$lim_{n \to \infty} \frac{1}{2 ^{n}} (\frac{1}{1 - \frac{1}{2 ^{n}}} + \frac{1}{1 - \frac{2}{2 ^{n}}} + \frac{1}{1 - \frac{3}{2 ^{n}}} + \dots + \frac{1}{1 - \frac{2 ^{n} - 1}{2 ^{n}}})$ is equal to
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Let $[t]$ denote the greatest integer less than or equal to $t$ . Then the value of the integral $\int_{- 3}^{101} ([sin (π x)] + e^{[c o s (2 π x)]}) dx$ is equal to
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Let $a_{n} = \int_{- 1}^{n} (1 + \frac{x}{2} + \frac{x ^{2}}{3} + \frac{x ^{3}}{4} + \dots + \frac{x ^{n - 1}}{n}) d x$ for $n \in N$ . Then the sum of all the elements of the set ${n \in N : a_{n} \in (2, 30)}$ is
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If $a = lim_{n \to \infty} \sum_{k = 1}^{n} \frac{2 n}{n ^{2} + k ^{2}}$ and $f (x) = \frac{1 - c o s x}{1 + c o s x}, x \in (0, 1)$ , then :
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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 ______.
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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 …
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Let $f : R \to R$ be continuous function satisfying $f (x) + f (x + k) = n$ , for all $x \in R$ where $k > 0$ and n is a positive integer. If …
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Let $f$ be a differentiable function in $(0, \frac{π}{2})$ . If $\int_{c o s x}^{1} t^{2} f (t) d t = sin^{3} x + cos x - 1$ , then $\frac{1}{3} f^{'} (\frac{1}{3})$ is equal to :
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The integral $\int_{0}^{1} \frac{1}{7 ^{[\frac{1}{x}]}} d x$ , where $[.]$ denotes the greatest integer function is equal to
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Let $[t]$ denote the greatest integer less than or equal to $t$ . Then, the value of the integral $\int_{0}^{1} [- 8 x^{2} + 6 x - 1] d x$ is equal to
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Let $f$ be a real valued continuous function on $[0, 1]$ and $f (x) = x + \int_{0}^{1} (x - t) f (t) d t$ . Then which of the following points $(x, y)$ lies on the curve $y = f (x)$ ?
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If $\int_{0}^{2} (2 x - 2 x - x^{2}) d x = \int_{0}^{1} (1 - 1 - y^{2} - \frac{y ^{2}}{2}) d y + \int_{1}^{2} (2 - \frac{y ^{2}}{2}) d y + I$ , then $I$ equals:
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The integral $\frac{24}{π} \int_{0}^{2} \frac{( 2 - x ^{2} ) d x}{( 2 + x ^{2} ) 4 + x ^{4}}$ is equal to ____.
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$\int_{0}^{5} cos (π (x - [\frac{x}{2}])) d x$ ,
Where [ $t$ ] denotes greatest integer less than or equal to $t$ , is equal to :
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The value of the integral $\frac{48}{π ^{4}} \int_{0}^{π} (\frac{3 π x ^{2}}{2} - x^{3}) \frac{s i n x}{1 + c o s ^{2} x} d x$ is equal to ____.
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The value of the integral $\int_{- 2}^{2} \frac{∣ x ^{3} + x ∣}{( e ^{x ∣ x ∣} + 1 )} d x$ is equal to :
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The integral $\int_{0}^{\frac{π}{2}} \frac{1}{3 + 2 s i n x + c o s x} d x$ is equal to:
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If $f (α) = \int_{1}^{α} \frac{l o g _{10} t}{1 + t} d t, α > 0$ , then $f (e^{3}) + f (e^{- 3})$ is equal to :
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Let $f (θ) = sin θ + \int_{- π /2}^{π /2} (sin θ + t cos θ) f (t) d t$ . Then the value of $\int_{0}^{π /2} f (θ) d θ$ is ______.
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The value of the integral $\int_{- π /2}^{π /2} \frac{d x}{( 1 + e ^{x} ) ( s i n ^{6} x + c o s ^{6} x )}$ is equal to
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If $b_{n} = \int_{0}^{\frac{π}{2}} \frac{c o s ^{2} n x}{s i n x} d x, n \in N$ , then
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The value of $b > 3$ for which
$12 \int_{3}^{b} \frac{1}{( x ^{2} - 1 ) ( x ^{2} - 4 )} d x = lo g_{e} (\frac{49}{40})$ , is equal to
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The value of $\int_{0}^{π} \frac{e ^{c o s x} s i n x}{( 1 + c o s ^{2} x ) ( e ^{c o s x} + e ^{- c o s x} )} dx$ is equal to
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The value of $\int_{\frac{- 1}{2}}^{\frac{1}{2}} ((\frac{x + 1}{x - 1})^{2} + (\frac{x - 1}{x + 1})^{2} - 2)^{1/2} d x$ is:
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If $\int_{- a}^{a} (∣ x ∣ + ∣ x - 2∣) d x = 22$ , $(a > 2)$ and $[x]$ denotes the greatest integer $\leq x$ , then $\int_{- a}^{a} (x + [x]) d x$ is equal to ______.
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If the value of the integral $\int_{0}^{5} \frac{x + [ x ]}{e ^{x - [x]}} d x = α e^{- 1} + β$ , where $α, β \in R$ , $5 α + 6 β = 0$ , and $[x]$ denotes the greatest integer less than or equal to $x$ ; then the value of $(α + β)^{2}$ is equal to
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The value of $\int_{- \frac{π}{2}}^{\frac{π}{2}} (\frac{1 + s i n ^{2} x}{1 + π ^{s i n x}}) d x$ is:
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Let a be a positive real numbers 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 :
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The value of the integral $\int_{- 1}^{1} lo g_{e} (1 - x + 1 + x) d x$ is equal to :
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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 $f (2) = e^{2}$ . Then the value of $\int_{0}^{2} f (x) d x$ is
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The value of the integral $\int_{1}^{3} [x^{2} - 2 x - 2] d x$ , where $[x]$ denotes the greatest integer less than or equal to $x$ , is
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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?
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If $[x]$ denotes the greatest integer less than or equal to $x$ , then the value of the integral $\int_{- π /2}^{π /2} [[x] - sin x] dx$ is equal to :
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The value of the integer $\int_{0}^{1} \frac{x d x}{( 1 + x ) ( 1 + 3 x ) ( 3 + x )}$ is:
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The value of $\int_{- 1}^{1} x^{2} e^{[x^{3}]} d x$ , where $[t]$ denotes the greatest integer $\leq t$ , is :
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If $\int_{0}^{100 π} e^{- (\frac{x}{π} - [\frac{x}{π}])} sin^{2} x d x = \frac{α π ^{3}}{1 + 4 π ^{2}}, α \in R$ , where $[x]$ is the greatest integer less than or equal to $x$ , then the value of $α$ is :
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