Definite Integration Questions (1340)

If f(x) = \lim_{n \to \infty} \frac{1 \cdot 2 + 2 \cdot 3 + \ldots + n(n+1)}{n^3}\left(\frac{a}{3}\right)^{1/2}\left(\frac{5 \cdot 3 \cdot 1}{5 \cdot 3 \cdot 1 \cdots (2n-1)}\right)^{1/2}\), then \int_0^1 f(x) \, d(x - [x])\) (where [\cdot] is G.I.F.) is

The value of the definite integral \int_0^{\pi/3} \ln(1 + \sqrt{3}\tan x) dx equals

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.]

The value of \(\sqrt{\pi\overset{2008}{\underset0{\int x\vert\sin\pi x\vert dx}}}\) is equal to:

Let \(f(x) = \lim_{n \to \infty} \dfrac{\cos x}{1 + (\tan^{-1} x)^n}\), then the value of \(\int_0^\infty f(x)\, dx\) is equal to:

If \(f(x)\) is a continuous function, then which of the following is true?

The value of \(\int_{1}^{a} [x] f'(x) \, dx\), \(a > 1\), where \([x]\) denotes the greatest integer not exceeding \(x\) is

If A = \(\left[\mathbf{a}_{i j}\right]_{n \times n}\), where aij = i100 + j100, then \(\lim _\limits{n \rightarrow \infty} \frac{\sum_\limits{i=1}^{n} a_{i i}}{n^{101}}\) equals :

If α, β are the roots of g(x) = ax2 + bx + c = 0 and f(x) is an even function, then the value of ∫−βα eg(x)/a · x · f(g(x))/g'(x) dx is equal to

Prove that $\int_{0}^{\pi/2} \frac{g(\sin x)}{g(\sin x) + g(\cos x)} dx = \int_{0}^{\pi/2} \frac{g(\cos x)}{g(\sin x) + g(\cos x)} dx = \frac{\pi}{4}$.

18. $\int \frac{4x^5 - 7x^4 + 8x^3 - 2x^2 + 4x - 7}{x^2(x^2 + 1)^2} dx$ equals

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

Let \(f : \mathbb{R} \to \mathbb{R}\) such that \(f(x + 2y) = f(x) + f(2y) + 4xy\) for all \(x, y \in \mathbb{R}\) and \(f(0) = 0\). If \(I_1 = \int_0^1 f(x) dx\), \(I_2 = \int_1^{3/2} f(x) dx\), and \(I_3 = \int_{1/2}^1 f(x) dx\), then

Let \(I_n = \int_{-1}^{1} x \left(1 + x + \frac{x^2}{2} + \frac{x^3}{3} + \ldots + \frac{x^{2n}}{2n}\right) dx\). If \(\lim_{n \to \infty} I_n\) can be expressed as \(\frac{p}{q}\) in its lowest form, then find the value of \(\frac{p}{q(p+q)}{10}\).

16. $\int \frac{dx}{(x-\alpha)\sqrt{(x-\alpha)(x-\beta)}}$ equals

The value of \(\int_{e^{2}}^{e^{4}} \frac{1}{x}\left(\frac{e^{\left(\left(\log _{e} x\right)^{2}+1\right)^{-1}}}{e^{\left(\left(\log _{e} x\right)^{2}+1\right)^{-1}}+e^{\left(\left(6-\log _{e} x\right)^{2}+1\right)^{-1}}}\right) d x\) is

If $I_n = \int_{0}^{\pi/4} \tan^n x \, dx$, then show that $I_n + I_{n-2} = \frac{1}{n-1}$

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

∫ \frac{5x^4 + 4x^5}{(x^5 + x + 1)^2} dx equals

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 \(3k(1)\).

705. Let \(I(n) = \displaystyle\int_0^{\pi} \ln(1 - 2n\cos x + n^2)\,dx\). Find the value of \(\dfrac{I(100)}{I(10)}\).

Let $\displaystyle\int\frac{2-\tan x}{3+\tan x}\,dx=\frac{1}{2}\!\left(\alpha x+\log_e|\beta\sin x+\gamma\cos x|\right)+C$, where $C$ is the constant of integration. Then $\alpha+\dfrac{\gamma}{\beta}$ is equal to:

Let F(x) be an indefinite integral of sin2x.Statement-1 : The function F(x) satisfies F(x + π) = F(x) for all real x.becauseStatement-2 : sin2(x + π) = sin2x for all real x.

Evaluate $\int \left( \frac{1}{8x+9} + e^{8x+9} \right) dx$

Find $\int [\cos(5+4x) + \sec^2(3-4x)] dx$

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

Evaluate $\int \frac{x^3 + x^2 + x + 1}{\sqrt{x^2 + 2x + 3}} dx$.

Find $\int \frac{1+x^2}{1+x^4} dx$

If ∫ 1 / ((x-1)^(4/3) * (x+3)^(5/3)) dx = A * ((αx - 1) / (βx + 3))^B + C, where C is the constant of integration, then the value of α + β + 20AB is ________.

If $f, g, h$ be continuous functions on $[0, a]$ such that $f(a - x) = -f(x)$, $g(a - x) = g(x)$ and $3h(x) - 4h(a - x) = 5$, then prove that $\int_{0}^{a} f(x)g(x)h(x) dx = 0$

The value of $\dfrac{8}{\pi}\displaystyle\int_0^{\pi/2}\dfrac{(\cos x)^{2023}}{(\sin x)^{2023}+(\cos x)^{2023}}\,dx$ is ___.

Let g(x) be an antiderivative for f(x). Then ln(1+(g(x))^2) is an antiderivative for

Evaluate $\int \frac{2x}{\sqrt{x^4 + 2x^2 + 4}} dx$

If $\displaystyle\int_{1/3}^4\log_e x\,dx=\dfrac{m}{n}\log_e\!\left(\dfrac{n^2}{e}\right)$, where $m$ and $n$ are coprime natural numbers, then $m^2+n^2-5$ is equal to ___.

The value of the integral $\displaystyle\int_1^2\left(\dfrac{t^4+1}{t^6+1}\right)dt$ is equal to:

Let a differentiable function $f$ satisfy $f(x)+\displaystyle\int_3^x\dfrac{f(t)}{t}\,dt=\sqrt{x+1}$, $x\geq3$. Then $12f(8)$ is equal to:

Let $\alpha\in(0,1)$ and $\beta=\log_e(1-\alpha)$. Let $P_n(x)=x+\dfrac{x^2}{2}+\dfrac{x^3}{3}+\cdots+\dfrac{x^n}{n}$, $x\in(0,1)$. Then the integral $\displaystyle\int_0^\alpha\dfrac{t^{50}}{1-t}\,dt$ is equal to:

∫ x2 + 3 / x6(x2 + 1) dx equals

If $\displaystyle\int_0^1(x^{21}+x^{14}+x^7)(2x^{14}+3x^7+6)^{1/7}\,dx=\dfrac{1}{l}(11)^{m/n}$, where $l,m,n\in\mathbb{N}$ and $m,n$ are coprime, then $l+m+n$ is equal to ___.

If $\displaystyle\int_0^\pi\dfrac{5^{\cos x}(1+\cos x\cos3x+\cos^2x+\cos^3x\cos3x)}{1+5^{\cos x}}\,dx=\dfrac{k\pi}{16}$, then $k$ is equal to ___.

Evaluate \(\displaystyle\lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^n\frac{1}{1+k/n}\) [JEE Main 2016]

Suppose J = ∫ \frac{\sin^2 x + \sin x}{1 + \sin x + \cos x} dx and K = ∫ \frac{\cos^2 x + \cos x}{1 + \sin x + \cos x} dx. If C is an arbitrary constant of integration then which of the following is/are correct?

$\int \frac{x^2 + x}{(e^x + x + 1)^2} dx$ equals

If $\displaystyle\sum_{i=1}^{n}(\sin^{-1} x_i + \cos^{-1} y_i) = \frac{9\pi}{...}$, evaluate $\displaystyle\int_{-1}^{1} x\ln(1+x^2)\cdot\frac{e^x}{1+e^{2x}}\,dx$

The value of the definite integral \[I = \int_0^{\pi/2} \frac{\cos^4 x + \sin x\cos^3 x + \sin^2 x\cos^2 x + \sin^3 x\cos x}{\sin^4 x + \cos^4 x + 2\sin x\cos^3 x + 2\sin^2 x\cos^2 x + 2\sin^3 x\cos x}\, dx\] is equal to \(\frac{\pi}{k}\). Find \(k\).

15. $\int [\sin \alpha \sin(x - \alpha) + \sin^2(\frac{x}{2} - \alpha)] dx$ equals

∫ln(x + √1 + x2) / √1 + x2 dx equals -

Let f(x) = \frac{2\sin^2 x - 1}{\cos x} + \frac{\cos x(2\sin x + 1)}{1 + \sin x} then \int e^x(f(x) + f'(x))dx (where c is the constant of integration)

Let \(f(x) = \begin{vmatrix} 2\cos 2x & \sin 2x & -\sin x \\ \sin 2x & 2\sin 2x & \cos x \\ \sin x & -\cos x & 0 \end{vmatrix}\), the value of \(\int_0^{\pi/2} \{f(x) + f'(x)\} dx\) is

The value of the definite integral \[\int_0^{10} \left((x-5) + (x-5)^2 + (x-5)^3\right) dx\]