Definite Integration Questions (1340)

Evaluate \(\displaystyle\int_0^1 x\,e^x\,dx\)

33. Suppose that a continuous function \(f(x)\) satisfies the relation \(\int_{x}^{x+1} f(t)\, dt = e^x\) for every \(x \geq 0\). The value of \(f(2) - f(0)\), equals:

The value of the integral $I = \int_0^\pi [\sin x + \cos x](\cos x - \sin x) dx$ is equal to (where $[.]$ denotes the greatest integer function)

Let \(f, g:(0, \infty) \rightarrow R\) be two functions defined by \(f(x)=\int_{-x}^{x}\left(|t|-t^{2}\right) e^{-t^{2}} d t\) and \(g(x)=\int_{0}^{x^{2}} t^{1 / 2} e^{-t} d t\). Then the value of \(\left(f\left(\sqrt{\log _{e} 9}\right)+g\left(\sqrt{\log _{e} 9}\right)\right)\)is equal to

If \int_0^{100} f(x) dx = a, then \sum_{r=1}^{100} \int_0^1 f(r-1+x) dx equals

Let f : [0, 1] \(\to\) R be a continuous function then the maximum value of \(\int_\limits{0}^{1} f(x) \cdot x^{2} d x-\int_\limits{0}^{1} x \cdot(f(x))^{2} d x\) for all such function(s) is :

284. \( L = \lim_{n \to \infty} \sqrt{n} \int_{0}^{1} \dfrac{dx}{(1+x^2)^n} \). Suppose that the above limit exists, then choose the correct option.

$\int \frac{x^2(1-\ln x)}{\ln^4 x - x^4} dx$ equals

If \(f(x)=\displaystyle\int_0^x t\sin(x-t)\,dt\), find \(f''(x)\). [JEE Main 2017]

\[\left|2\left(x^2 + \frac{1}{x^2}\right) + |1 - x^2|\right| = 4\left(\frac{3}{2} - 2^{x^2 - 3} - \frac{1}{2^{x^2+1}}\right)\]If \(x_1\) and \(x_2\), where \(x_1 [Note: \(|\cdot|\) denotes the absolute value function, \(\{\cdot\}\) denotes the fraction part function, \([\cdot]\) denotes the floor function]

Let \(a, b\) and \(c\) be non-zero real numbers such that \(\int_0^3 (3ax^2 + 2bx + c) dx = \int_1^3 (3ax^2 + 2bx + c) dx\), then

Evaluate \(\int_0^{\pi/2} \frac{\sin x}{\sin x + \cos x} dx\)

The function f(x) is defined as follows: \(f(x) = x^2,\quad -1 \le x \(f(x) = \sqrt{x},\quad 1 \le x \(f(x) = \sqrt{2},\quad 2 \le x \le 4\) Evaluate \(\int_{-1}^{4} f(x)\,dx\).

∫₋₁⁰ (3^{2[x]} - x⁴)/(3^{2[x]} - [x]²)dx is equal to (where [·] denotes greatest integer function):

Let a function f: ℝ → ℝ be defined as f(x) = x + sin x. The value of ∫₀^{2π} f⁻¹(x)dx will be:

The value of $\int_0^{\pi} \frac{x\sin x}{1+x^2}dx$ is equal to

Let \( I = \int_0^{\pi/2} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} \, dx \). Find the value of \(I\).

888. Let \(k(x)\) be a continuous function satisfying the equation \(\displaystyle\int_0^{x^3} k(t)\, dt = x^{1+x^2}\), find the value of \(3k(1)\).

$\int_{0}^{\pi/2} \frac{2^{\sin x}}{2^{\sin x} + 2^{\cos x}} dx$ equals -

Evaluate: \(\displaystyle\int_0^{5\pi/12} [\tan x]\,dx\), where [·] is the greatest integer function (up to four decimal places).

The value of $\lim_{n \to \infty} \sum_{k=1}^{n} \frac{n^{\frac{1}{6}}}{\sqrt[6]{k^4 + n^4}}$ is equal to

If \(f\) and \(g\) are two functions such that \(2f(1) = g(2) = 4\) and \(2f(9) = g(10) = 20\) and \(\int_0^2 (x^2 g(f(x^3+1))f'(x^3+1) - 3x^2)\,dx = 0\), then find the value of \(\int_4^{20} g^{-1}(x)\,dx\).

If \int 2 \pi 96x 2 cos 2 x dx = \pi (\alpha\pi 2 + \beta) , \alpha, \beta \in Z , then (\alpha + \beta) equals 2 - x (1+e ) 2

Let $f(x) = \displaystyle\int_0^x t(t^2-9t+20)\,dt$, $1\leq x\leq 5$. If the range of $f$ is $[\alpha,\beta]$, then $4(\alpha+\beta)$ equals:

Let f(x) = \frac{x}{(1+nx^n)^{1/n}} for n \ge 2 and g(x) = \underbrace{(f \circ f \circ \dots \circ f)}_{f \text{ occurs } n \text{ times}}(x). Then \int x^{n-2} g(x) dx \text{ equals.}

Let f : R \to R be a twice differentiable function such that f (2) = 1. If F(x) = xf (x) for all x \in R, and \int , then F (2) + \int is equal to : 2 ′ 2 ′′ ′ 2 2 \int x F (x)dx = 6 x F (x)dx = 40 F(x)dx 0 0 0

Evaluate: $\int_{-10}^{20} [\cot^{-1} x] dx$. Here $[.]$ is the greatest integer function.

13. ∫(sin(101x)·sin⁹⁹x)dx equals

Let for $f(x) = 7\tan^8 x + 7\tan^6 x - 3\tan^4 x - 3\tan^2 x$, $I_1 = \displaystyle\int_0^{\pi/4}f(x)\,dx$ and $I_2 = \displaystyle\int_0^{\pi/4}xf(x)\,dx$. Then $7I_1+12I_2$ is equal to:

Find the following limit: 15. \(\lim_{n \to \infty} \left[\left(1+\frac{1}{n}\right)\left(1+\frac{2}{n}\right)\left(1+\frac{3}{n}\right)\cdots\left(1+\frac{n}{n}\right)\right]^{1/n}\)

The value of $\int_0^1 \left[\sin^{-1}\left(\frac{2x}{1+x^2}\right)\right]dx$ is equal to

If $I = \displaystyle\int_0^{\pi}\frac{\sin^{3/2}x}{\sin^{3/2}x+\cos^{3/2}x}\,dx$, then $\displaystyle\int_0^{2I}\frac{x\sin x\cos x}{\sin^4 x+\cos^4 x}\,dx$ equals:

If $\displaystyle\int_{-\pi/2}^{\pi/2}\frac{96x^2\cos^2 x}{1+e^x}\,dx = \pi(\alpha\pi^2+\beta)$, $\alpha,\beta\in\mathbb{Z}$, then $(\alpha+\beta)^2$ equals

If $I(m,n) = \displaystyle\int_0^1 x^{m-1}(1-x)^{n-1}\,dx$, $m,n>0$, then $I(9,14)+I(10,13)$ is

If $\int_0^{\pi/4} x\cos\left(\frac{b}{1+x^2}\right)dx = k\int_0^{b} \frac{1+u^2}{1-u^2}du$, then the value of $k$ is equal to

∫ \frac{1-x^7}{x(1+x^7)} dx \text{ equals -}

Let $[t]$ denote the greatest integer $\leq t$. Then $\dfrac{2}{\pi}\displaystyle\int_{\pi/6}^{5\pi/6}(8[\csc x]-5[\cot x])\,dx$ is equal to _______

Find \(\displaystyle\int_0^1 x^n(1-x)^n\,dx\) [JEE Main 2021]

The value of the integral $I = \int_{\pi}^{\pi} (|\sin x| + |\cos x|)dx$ (where $[\cdot]$ denotes the greatest integer function) is equal to

Let f (x) = \int x 0 t -8t+15 t dt, x \in R . Then the numbers of local maximum and local minimum points of f , e respectively, are :

Evaluate the following: 25. \(\int_0^{\pi/2} \cos 2x \cdot \log(1 + \tan x) \, dx\)

Find $I = \int_0^1 \frac{\sqrt{1-x}}{\sqrt{x} + \sqrt{1-x}} dx$ (rationalising the denominator)

The value of $\int_{-\pi}^{\pi}\dfrac{2y(1+\sin y)}{1+\cos^2 y}\,dy$ is:

Let $f:[0,2]\to\mathbb{R}$ be defined as $f(x)=\begin{cases}e^{\min\{x^2,x-[x]\}}, & x\in[0,1)\\e^{[x-\log_e x]}, & x\in[1,2]\end{cases}$. Then $\displaystyle\int_0^2 x f(x)\,dx$ is

If $24\displaystyle\int_0^{\pi/6}\left(\sin\!\left|4x-\frac{\pi}{12}\right|+\left[2\sin x\right]\right)dx = 2\pi+\alpha$, where $[\cdot]$ denotes the greatest integer function, then $\alpha$ is equal to ____.

If $f:\mathbb{R}\to\mathbb{R}$ be a continuous function satisfying $\displaystyle\int_0^{\pi/2}f(\sin 2x)\sin x\,dx+\alpha\int_0^{\pi/4}f(\cos 2x)\cos x\,dx=0$, then the value of $\alpha$ is

280. The value of the definite integral \[ \int_{1/3}^{1} \frac{\pi\cos\!\left(\dfrac{2\pi}{3}x\right) + \pi\cos\!\left(\dfrac{\pi}{3}x\right)}{\sin\!\left(\dfrac{\pi}{2}x\right)\sin\!\left(\dfrac{2\pi}{3}x\right) + 2\sin\!\left(\dfrac{\pi}{2}x\right)\sin\!\left(\dfrac{\pi}{3}x\right)}\, dx \] is equal to:

Let $I_1 = \int_0^e x^2dx$ and $I_2 = \int_0^e 2x e^x dx$, then the value of $I_1 + I_2$ is equal to

Let $f$ be a real valued continuous function defined on the positive real axis such that $g(x) = \displaystyle\int_0^x tf(t)\,dt$. If $g(x^3) = x^6+x^7$, then value of $\displaystyle\sum_{r=1}^{15}f(r^3)$ is:

Let $f(x) = \displaystyle\int_0^{x^2}\frac{t^2-8t+15}{e^t}\,dt$, $x\in\mathbb{R}$. Then the numbers of local maximum and local minimum points of $f$, respectively, are: