Definite Integration Questions - JEE

$\lim _\limits{\lambda \rightarrow 0}\left(\int_{0}^{1}(1+x)^{\lambda} d x\right)^{1 / \lambda}$ is equal to:

The minimum value of the twice differentiable function f(x) = $\int \limits_0^x e^{x-t} f^{\prime}(t) d t-\left(x^2-x+1\right) e^x$, $ x \in R$, is:

The maximum value of f(x) = $\int_\limits{0}^{1}$ t sin(x + $\pi$t) dt, is

The area enclosed by the curve y2 + x4 = x2 is :

If the integral $\int \frac{5\tan x}{\tan x - 2}dx = x + a\ln|\sin x - 2\cos x| + k$, then a is equal to

$\int \frac{dx}{x(x^n+1)}$ is equal to

Evaluate: $ I = \int_{-\pi/2}^{\pi/2} \dfrac{\sin^2 x}{1+2^x} \, dx $(1) $\dfrac{\pi}{8}$ (2) $\dfrac{\pi}{2}$ (3) $\dfrac{\pi}{4}$ (4) $\dfrac{\pi}{6}$

If $\lim _\limits{n \rightarrow \infty} \frac{1^{a}+2^{a}+\ldots+n^{a}}{(n+1)^{a-1}[(n a+2)+\ldots(n a+n)]}=\frac{1}{60}$ for some positive real number a, then a is equal to:

Let $f:[0,5] \to \mathbb{R}$ be such that $f''(x) = f''(5-x)$, $\forall\, x \in [0,5]$, $f'(0) = 1$ and $f'(5) = 7$, then the value of $\displaystyle\int_1^4 f'(x)\, dx$ is:

Let 2a > -1. If the area of the region of the plane defined by {(x, y) : x $\geq$ 0, 2y - x $\geq$ 0, ax + y - 3 $\leq$ 0} is equal to 3, then the value of a, lies in :

The value of $\displaystyle\int_0^1 x(1-x)^{99}\,dx$ is ______ $\times 10^{-5}$.

The value of $\displaystyle\int_{-5}^{5} |x + 2|\,dx$ is ______.

Given $f$ and $g$ are two continuous functions such that $f(x) = f(a - x)$ and $g(x) + g(a - x) = 4$. Now, $I = \int_0^a f(x)g(x)\,dx$. Then $I$ equals:

If $\int_\limits0^x$ f(t) dt = x2 + $\int_\limits x^1$ t2f(t) dt, then $f^{\prime}(\frac 1 2)$ is:

Given that $g(x) = \int_0^x \cos 4t\,dt$, which of the following is true?

Let $\beta(m, n)=\int_{0}^{1} x^{m-1}(1-x)^{n-1} d x, m, n \gt 0$. If $\int_{0}^{1}\left(1-x^{10}\right)^{20} d x=a \times \beta(b, c)$ then $100(a+b+x)$ equals ________.

The value of $k \in N$ for which the integral $I_{n}=\int_{0}^{1}\left(1-x^{k}\right)^{n} d x, n \in \mathbb{N}$, satisfies $147 I_{20}=148 I_{21}$ is:

Let $I_n = \int_0^{\pi/4} \tan^n x\,dx$. Find $\lim_{n \to \infty} n(I_n + I_{n+2})$.

$\int_0^{10\pi} |\sin x|\, dx$ is

Let d1 [(x1, y1), (x2, y2)] = |x1 - x2| + |y1 - y2| and d2 [(x1, y1), (x2, y2)] = $\sqrt{\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}}$ denote the distance between (x1, y1) and (x2, y2) on the coordinate plane. The area of the region enclosed by the set of points (x, y) satisfying d1 [(x, y), (0, 0)] $\geq$ 1 and d2 [(x, y), (0, 0)] $\leq$ 1, is :

The value of $\int_{-\pi}^{\pi} \dfrac{\cos^2 x}{1 + a^x}\, dx,\, a > 0$, is

The integral \int_{\pi/6}^{\pi/3} \sec^{2/3} x \csc^{4/3} x dx is equal to

The value of integral $\displaystyle\int_{\pi/4}^{3\pi/4} \frac{x}{1+\sin x}\, dx$ is

Evaluate: \[I = \int_{-1.5}^{3.5} |x-1|\, dx\]

Let \[ I = \int_{0}^{7} \frac{x\log x}{(1+x^2)^2}\, dx \] Find the value of $I$.

The integral $\int_{0}^{\pi} \frac{8 x d x}{4 \cos ^{2} x+\sin ^{2} x}$ is equal to

Write the given integration $I = \int \dfrac{dx}{x^{22}(x^7 - 6)}$ in the form involving substitution $p = 1 - \dfrac{6}{x^7}$. The answer is $\dfrac{1}{(42)(216)}\int \dfrac{1-p^3 - 3p + 3p^2}{p}\,dp$. Find the integer value of the denominator constant (i.e., $54432 = 42 \times 216 \times k$ gives $k$).

Here, $F(x^2) = x^4 + x^5 = \int_0^{x^2} t\, f(t)\, dt$. Find $\sum_{r=1}^{12} f(r^2)$.

The function g(x) is

$\int \frac{\ln 2x}{4x}$ dx is equal to

Evaluate $\displaystyle\lim_{n\to\infty}\sum_{k=1}^n\frac{1}{\sqrt{n(n+k)}}$ [JEE Main 2019]

Let $f(x) = \int_0^x e^{x-y} f'(y) dy - (x^2 - x + 1)e^x$. Find the number of roots of the equation $f(x) = 0$.

Evaluate $\displaystyle\int_0^{\pi/2}\frac{\sin x}{\sin x+\cos x}\,dx$ [JEE Main 2016]

Evaluate $I = \displaystyle\int_{e^{\pi/6}}^{e^{\pi/2}} \frac{\sin(\ln(\sin(\ln x)))\cdot\cos(\ln x)}{x\sin(\ln x)}\,dx$. Find $\cos^{-1}(I+1)$.

If x(x4 + 1)f(x) = 1, then $\int\limits_1^2 {f(x)} $dx equals

Evaluate $\displaystyle\int_0^{\pi/2}\sin^4 x\,dx$ [JEE Main 2020]

The value of the definite integral $\int_\limits{-1}^{1} e^{-x^{4}}\left(1+\ln (x+\sqrt{x^{2}+1})+5 x^{3}-4 x^{4}\right) d x$ is equal to

If $\int_{0}^{1} \frac{1}{\sqrt{3+x}+\sqrt{1+x}} d x=a+b \sqrt{2}+c \sqrt{3}$ where $a, b, c$ are rational numbers, then $2 a+$ $3 b-4 c$ is equal to:

The area bounded by y = 2 - |2 - x| and $y=\frac{3}{|x|}$ is :

Evaluate $\displaystyle\int_0^1\sqrt{\frac{x}{1+x}}\,dx$ [JEE Main 2020]

Definite Integration Questions (1340)