Definite Integration Questions - JEE

If $\int_0^\pi \sqrt{(\cos x + \cos 2x + \cos 3x)^2 + (\sin x + \sin 2x + \sin 3x)^2} dx$ has the value equal to $\left(\frac{\pi}{k} + \sqrt{w}\right)$ where $k$ and $w$ are positive integers then $k^2 + w^2 = $ ____.

If the absolute value of the integral $I = \int_{\pi/4}^{\pi/2} \frac{x \cdot \cos 2x \cdot \cos x}{\sin^3 x} dx$ in the lowest form is $-\frac{a}{b}$ where $a,b \in \mathbb{N}$, then $(a+b) = $ ____.

If $\int_0^1 \frac{c\sin^3 x}{4-\cos^2 x} dx = \pi\left[1-\frac{a\ln b}{c}\right]$ where $a$ and $b$ are prime and $c \in \mathbb{N}$, then $a+b+c = $ ____.

Consider a real valued continuous function $f$ such that $f(x) = \sin x + \int_{-\pi/2}^{\pi/2} (\sin x + tf(t)) dt$, then minimum value of $f(x)$ is____.

Evaluate $I = \int_{\pi/6}^{\pi/3} \dfrac{dx}{1 + \sqrt{\tan x}}$.

Evaluate $\int_{0}^{100} (x - [x])\,dx$.

The value of $\int_1^8 x\sin[x^2 - \pi] dx$, where $[.]$ denotes the greatest integer function is:

Evaluate $\displaystyle\int_0^{\pi/4}\sec^3 x\,dx$ [JEE Main 2019]

Area bounded by the curves $y = \left[\frac{x^2}{64} + 2\right]$ ([$.$] denotes the greatest integer function), $y = x - 1$ and $x = 0$ above the $x$-axis is:

The value of $\int_0^1 |1 - x^2| dx$ is

The value of $\int_1^a \frac{x^a - 1}{\log x} dx$ is:

As we have $f'(x) = f(x)$, $f(0) = 1$ and $f(x) + g(x) = x^2$. Then $\displaystyle\int_0^1 f(x)\cdot g(x)\,dx$ equals:

If $P = \int_0^\infty \frac{x^2}{1+x^4} dx$; $Q = \int_0^\infty \frac{xdx}{1+x^4}$ and $R = \int_0^\infty \frac{dx}{1+x^4}$, then:

If $\int_0^2 \frac{\ln(1+2x)}{1+x^2} dx = (\tan^{-1}a)(\ln\sqrt{b})$ where $a,b \in \mathbb{N}$, then:

If $I = \int_3^4 \frac{1}{\sqrt[3]{\ln x}} dx$, then:

If the value of $\lim_{n \to \infty} \left(n^{-3/2}\right) \sum_{j=1}^{\sqrt{n}} \sqrt{j}$ is equal to $\sqrt{N}$, then the value of $N/12$ is ____.

Let $J = \int_{-5}^4 (3-x^2)\tan(3-x^2) dx$ and $K = \int_{-2}^1 (6-6x+x^2)\tan(6x-x^2-6)dx$. Then $J + K$ is ____.

Let $I_n = \int_0^1 x^n\sqrt{1-x^2} dx$ then find the value of $\lim_{n \to \infty} \frac{I_n}{I_{n-2}}$.

For differentiable function $f(x)$, if $\int_0^a f'(x)\left([x]-x+\frac{1}{2}\right) dx = A_1 \int_0^a f(x) dx + A_2 f(0) + A_3 f(a) + A_4 \sum_{r=0}^{a} f(r)$, (where $[.]$ Denotes the G.I.F and $A_1, A_2, A_3, A_4$ are constant $n \in \mathbb{N}$) then $A_1 + A_2 + A_3 + A_4$ is equal to ____.

For positive integers $k = 1, 2, 3, \ldots, n$, let $S_k$ denotes the area of $\triangle AOB_k$ (where 'O' is origin) such that $\angle AOB_k = \frac{k\pi}{2n}$, $OA = 1$ and $OB_k = k$. If the value of $\lim_{n \to \infty} \frac{1}{n^2} \sum_{k=1}^n S_k = \frac{a}{\pi^2}$, then 'a' is equal to

If $\lim_{n \to \infty} \sum_{k=1}^n \frac{{^n}C_k}{n^k(k+3)} = p$ then $p$ is ____.

Let $f(x)$ be a continuous function with continuous first derivative on $(a, b)$, where $b > a$, and let $\lim_{x \to a^+} f(x) = \infty$, $\lim_{x \to b^-} f(x) = -\infty$ and $f'(x) + f^2(x) \geq -1$, for all $x$ in $(a, b)$, if the minimum value of $(b-a)$ equals to $k$ then $k$ is ____.

Let $y = f(x)$ be a quadratic function with $f'(2) = 1$. Then the value of the integral $\int_{2-\pi}^{2+\pi} f(x) \sin\left(\frac{x-2}{2}\right) dx$ is ____.

If $\int_0^{\ln t} \frac{t \ln 2}{x^2+t^2} dt = \frac{\pi \ln 2}{4}$ $(x>0)$ then the number of integral values of $'x'$ satisfying this equation is____.

Let $F(x) = \int_{-1}^x \sqrt{4+t^2} dt$ and $G(x) = \int_x^1 \sqrt{4+t^2} dt$ then the value of $(FG)'(0)$ is____ (where dash denotes the derivative).

If $\displaystyle\int_0^x f(t)\, dt = x^2 + \int_x^1 t^2 f(t)\, dt$, then $f'(1/2)$ is:

Evaluate the definite integral: \[I = \int_0^1 \frac{e^x + e^{-x}}{\sqrt{11 - e^{2x} - e^{-2x}}}\, dx\]

If $f(x)$ and $g(x)$ are both continuous functions then the value of \[\displaystyle\int_{\ln\lambda}^{\ln(1/\lambda)} \dfrac{f\!\left(\dfrac{x^2}{4}\right)(f(x) - f(-x))}{g\!\left(\dfrac{x^2}{4}\right)(g(x) + g(-x))}\, dx\] is equal to:

If $\int x^5 e^{-x^2} dx = g(x)e^{-x^2} + C$, where $C$ is a constant of integration, then $g(-1)$ is equal to

Find the value of $ \int_{-5}^{5} |x + 2| \, dx $.

The value of $\int_0^{\pi} [\cos x]\,dx$ (where $[\cdot]$ denotes the greatest integer function) is:

If $\int 4x^3 e^{-4x^3} dx = \frac{1}{48}e^{-4x^3}f(x) + C$, where $C$ is a constant of integration, then $f(x)$ is equal to

The integral $\int \frac{\sin^2 x\cos^2 x}{(\sin^3 x+\cos^3 x)^2}dx$ equal to

The value of definite integral $ \displaystyle\int_{-\pi}^{\pi} \dfrac{2x(1 + \sin x)}{1 + \cos^2 x}\, dx $ is:

Suppose that f is a function on the interval [1, 3] such that $-1 \leq f(x) \leq 1$ for all x and $\int_1^3 f(x)\,dx = 0$. Find the maximum value of $\int_1^3 \dfrac{f(x)}{x}\,dx$ be?

We have \[I = \int \frac{\sin x}{\sin(x - \alpha)} dx\] If $I = Ax + B\ln|\sin(x-\alpha)| + C$, then find the values of $A$ and $B$.

Given $I = \int \dfrac{3x^{13} + 2x^{11}}{(2x^4 + 3x^2 + 1)^4}\,dx$. Then $I$ equals:

The integral $\displaystyle\int_{\pi/6}^{\pi/4} \frac{dx}{\sin 2x(\tan^5 x + \cot^5 x)}$ equals:

The integral $\displaystyle\int_0^{\pi/2}\sqrt{1+4\sin^2\dfrac{x}{2}-4\sin\dfrac{x}{2}}\,dx$ equals

The value of $\displaystyle\int_0^{2\pi} [\sin 2x(1 + \cos 3x)]\,dx$, where [t] denotes the greatest integer function, is:

$\lim_{x \to \infty} \sum_{r=1}^{n} \dfrac{1}{n} e^{r/n}$ is

$\int_0^{\pi} x f(\sin x)\, dx$ is equal to

[JEE Main 2019] $\displaystyle\int\frac{2\sin x}{2+\sin 2x}\,dx$ equals (where $C$ is a constant)

If $\displaystyle\int_0^x f(t)\,dt = e^x - ae^{2x}\int_0^1 f(t)e^{-t}\,dt$, then $f(1)+2f(2)$ is equal to:

If $f(x)$ is a differentiable function defined for all positive real numbers such that $xf(x) = x + \displaystyle\int_1^x f(t)\,dt$, then the value of $\displaystyle\sum_{k=1}^{10} f(e^k)$ is:

For $n \geq 1$, Let $G_n$ be the geometric mean of $\left\{\sin\frac{k\pi}{2n} : 1 \leq k \leq n\right\}$, then $\lim_{n\to\infty} G_n$ equals:[Note: [k] denotes greatest integer function less than or equal to k.]

Evaluate $\displaystyle\int_0^{1/2}\frac{dx}{\sqrt{1-x^2}}$ [JEE Main 2015]

The value of $\int_1^2 \frac{1}{1 + \log x} dx$ is

Let $g(x) = \int_0^x f(t) dt$, where f is such that $1 \leq f(t) \leq 2$ for $t \in [0,1]$ and $0 \leq f(t) \leq 1/2$ for $t \in (1,2]$. Then $g(2)$ satisfies the inequality

The value of $\int_0^1 (1 + \log x)\, dx$ is

Definite Integration Questions (1340)