Definite Integration Questions - JEE

If $I = \displaystyle\int_{-\frac{\sqrt{3}}{2}}^{\frac{\sqrt{3}}{2}} \sqrt{\frac{1-x}{1+x}}\,\sin^{-1}x\,dx = \frac{\pi}{M} - \sqrt{N}$, find the value of $(M+N)$.

Evaluate: $ L = \lim_{n \to \infty} \sum_{k=0}^{n-1} \frac{k}{n} \left[ \left(\frac{k+1}{n}\right)^{\frac{1}{m}} - \left(\frac{k}{n}\right)^{\frac{1}{m}} \right] $. Find the value of $m$ if $L = \dfrac{1}{10}$.

The value of $\displaystyle\int_0^{100\pi} \left(\left[\cot^{-1} x\right] + \left[\tan^{-1} x\right]\right) dx$ equals ________. (where $[\cdot]$ denotes the greatest integer function)

Let $ f(x) $ be a continuous function $ \forall\, x \in R $ such that \[ \lim_{x \to \pi/4} \frac{\displaystyle\int_{2}^{\sec^2 x} f(t)\,dt}{x^2 - \dfrac{\pi^2}{16}} = \frac{k}{\pi} f(a) \] where $ a, k \in N $, then the value of $ k^a $ is equal to:

$\lim_{n \to \infty} \left[\dfrac{1}{n^2} \sec^2 \dfrac{1}{n^2} + \dfrac{2}{n^2} \sec^2 \dfrac{4}{n^2} + \cdots + \dfrac{1}{n^2} \sec^2 1\right]$ equals

The integral $\int\left(1+x-\frac{1}{x}\right)e^{x+\frac{1}{x}}dx$ is equal to

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:

Find the following limit: 16. $\lim_{n \to \infty} \left[\left(1+\frac{1}{n^2}\right)\left(1+\frac{2^2}{n^2}\right)\left(1+\frac{3^2}{n^2}\right)\cdots\left(1+\frac{n^2}{n^2}\right)\right]^{1/n}$

Find $\int \frac{dx}{x^2 + a^2}$

[JEE Main 2019] $\displaystyle\int\frac{dx}{x(x^n+1)}$ equals

Evaluate $$\int \text{cosec} x \ln |\cot x - \text{cosec} x| dx$$

Evaluate the integral: \[ I = \int \frac{3(\tan x - 1)\sec^2 x}{(\tan x + 1)\sqrt{\tan^3 x + \tan^2 x + \tan x}}\, dx \]

If $L = \lim_{n \to \infty}\left(\dfrac{1}{\sqrt{n}\sqrt{n+1}} + \dfrac{1}{\sqrt{n}\sqrt{n+2}} + \cdots + \dfrac{1}{\sqrt{n}\sqrt{n+n}}\right) = a\sqrt{b} - c$ and $a^4 + b^3 + c^2 + d = 29$, find $d$.

Evaluate $\int \frac{dx}{\sqrt{9 - 16x^2}}$

If $\int e^{\sec x}(\sec x \tan x + (\sec x \tan x + \sec^2 x))\,dx = e^{\sec x} f(x) + C$, then a possible choice of $f(x)$ is

Given\[ \int \frac{dx}{x^3(1+x^6)^{2/3}} = x\, f(x)\cdot(1+x^6)^{1/3} + C \]find $f(x)$.

If In = ∫(sinx)ⁿ dx where n ∈ ℕ, then 5I4 - 6I6 is equal to:

The value of definite integral $\displaystyle\int_{1}^{\sqrt{3}} \left(x^{2x^2+1} + \ln\left(x^{x^{(2x^2+1)}}\right)\right)dx$ is equal to:

If $f'(x) + g'(x) = (f(x) + g(x))^2 + 1$, then adding two given equations gives $\displaystyle\int \dfrac{f'(x)+g'(x)}{(f(x)+g(x))^2+1}\,dx = \displaystyle\int 1\,dx$. What is the result?

If $\int x^{26}(x-1)^{17}(5x-3)\, dx = \dfrac{x^{27}(x-1)^{18}}{k} + C$, where $C$ is constant of integration, then the value of $k$ is:

Let $f(x) = \displaystyle\int \dfrac{2x}{(x^2+1)(x^2+3)}\,dx$. If $f(3) = \dfrac{1}{2}(\log_e 5 - \log_e 6)$, then $f(4)$ is equal to

If $I_{m,n} = \int \cos^m x \sin nx dx$, then $7I_{4,3} - 4I_{3,2} =$$

Evaluate $\int \frac{dx}{e^x \sec x(1 + \tan x)}$

If $\int \dfrac{2x+5}{\sqrt{7-6x-x^2}}\, dx = A\sqrt{7-6x-x^2} + B\sin^{-1}\left(\dfrac{x+3}{4}\right) + C$, where $C$ is a constant of integration, then the values of $A$ and $B$ are:

Evaluate the integral: \[I = \int \frac{\sin^2 x \cos^2 x}{(\sin^3 x + \cos^3 x)^2} dx\]

If $ f(x) = \displaystyle\int \frac{3x^2 - x^{-2}}{\left(x^3 + 1 + \dfrac{1}{x}\right)^2}\, dx $ and $ f(0) = 0 $, then $ f(-1) $ equals:

$\displaystyle\int\frac{3x+1}{(x+1)^2(x^2+1)}\,dx$ equals (where $C$ is the constant of integration)

Let $f(xy) = f(x) \cdot f(y)$, $\forall x > 0, y > 0$ and $f(1 + x) = 1 + x[1 + g(x)]$, where $\lim_{x \to 0} g(x) = 0$, then $$\int \frac{f(x)}{f'(x)} dx$$ is:

[JEE Main 2023] $\displaystyle\int\frac{3x^2+2}{(x^2+x+1)^2}\,dx$ equals (where $C$ is a constant)

If $f(x) = \displaystyle\int \dfrac{(3x^4 - 1)}{(x^4 + x + 1)^2}\, dx$ and $f(0) = 0$, then $f(-1)$ is equal to:

For $x > 0$, let $f(x) = \int_{1}^{x} \frac{\log t}{1+t} dt$. Then $f(x) + f\left(\frac{1}{x}\right)$ is equal to:

The value of the definite integral $\displaystyle\int_{1}^{\sqrt{3}}\left(x^{2x^2+1}+\ln\left(x^{x^{\left(2x^2+1\right)}}\right)\right)dx$ is equal to:

$\displaystyle\int \frac{x^2-1}{(x+1)\sqrt{x^2+x}}\,dx$ equals (where $C$ is constant of integration)

Find $\int \frac{\sin x}{1 + \sin x \cos x} dx$

Evaluate: $\displaystyle\int_{-4}^{4} \frac{x^2}{(x^2+16)(1+e^x)}\,dx$ (up to four decimal places).

If $\displaystyle\int(\sin x)^{-11/2}(\cos x)^{-5/2}\,dx=-\dfrac{p_1}{q_1}(\cot x)^{5/2}-\dfrac{p_2}{q_2}(\cot x)^{-5/2}-\dfrac{p_3}{q_3}(\cot x)^{1/2}+\dfrac{p_4}{q_4}(\cot x)^{-3/2}+C$, where $p_i$ and $q_i$ are positive integers with $\gcd(p_i,q_i)=1$ for $i=1,2,3,4$ and $C$ is the constant of integration, then $\dfrac{15p_1p_2p_3p_4}{q_1q_2q_3q_4}$ is equal to

Evaluate $\int (2x-4)\sqrt{4+3x-x^2} dx$

Let I_n = ∫₁^e (ln x)^n d(x²). Then the value of 2I_n + nI_{n-1} equals:

77. $\int \cos(\log_e x) dx$ is equal to

Evaluate the integral: \[I = \int \frac{x^{5m-1} + 2x^{4m-1}}{(x^{2m} + x^m + 1)^3} dx\]

Evaluate $\int \frac{2x - \sqrt{\arcsin x}}{\sqrt{1-x^2}} dx$

The value of $\sqrt{2}\int \frac{\sin x\, dx}{\sin\left(x - \frac{\pi}{4}\right)}$ is

If $I_1 = \int_{1}^{3} f(x^3 - 2x^2 - 5x + 2020)\,dx$ and $I_2 = \int_{1}^{3} f(x^3 - 2x^2 - 5x + 2020)\,dx$ are related such that $\dfrac{2I_1}{3I_2} = \dfrac{a}{b}$, find the value of $a + b$.

Given \[I = \int \frac{(\sin^n\theta - \sin\theta)^{\frac{1}{n}}\cos\theta}{\sin^{n+1}\theta}\,d\theta\] Evaluate the integral.

Let $I = \displaystyle\int (\sin 4x)\, e^{\tan^2 x}\,dx$. If $I = A^{10}$ when evaluated appropriately, find the value of $\dfrac{A^{10}}{10}$ (in decimal). Given that the answer is of the form $A = -2\cos^4 x \cdot e^{\tan^2 x} + C$, and $\dfrac{A^{10}}{10} = \dfrac{1024}{100}$.

$\displaystyle\int \cos(100x)\cdot\sin^{95}x\,dx$ equals

[JEE Main 2023] If $\displaystyle\int\frac{x}{1+x^3}\,dx = p\ln|1+x|+q\ln|1-x+x^2|+r\tan^{-1}\!\dfrac{2x-1}{\sqrt3}+C$, then the value of $18(p+q+r)$ is

Evaluate $\displaystyle\int_{-1}^{1.5} [x^2]\, dx$, where $[x]$ is the greatest integral (floor) function.

Evaluate $$\int \frac{\cos^4 x \, dx}{\sin^3 x \{\sin^5 x + \cos^5 x\}^{3/5}}$$

If f_n(x) = \frac{n+1}{(n+1)!} \cdot x^n, then the value of \int_1^n \frac{1}{f_n(x)} \, dx is

Definite Integration Questions (1340)