Definite Integration

The value of the definite integral
$$\int_0^2\frac1{3^x+3}\,dx$$
is

If $\alpha$ is the area of the common region that lies inside both the given ellipses, then the value of $\cot\alpha$ is ________.

Let $[\cdot]$ denote the greatest integer function. Then the value of $\displaystyle\int_0^3\left(\frac{e^x+e^{-x}}{[x]!}\right)dx$ is :

If $\alpha=\displaystyle\int_0^{2\sqrt3}\log_2\left(x^2+4\right)dx+\int_2^4\sqrt{2^x-4}\,dx$, then $\alpha^2$ is equal to ________.

The value of $\displaystyle\int_0^{20\pi}\left(\sin^4x+\cos^4x\right)dx$ is equal to :

Let $f$ be a real polynomial of degree $n$ such that $f(x)=f'(x)f''(x)$, for all $x\in\mathbb R$. If $f(0)=0$, then $36\left(f'(2)+f''(2)+\displaystyle\int_0^2f(x)\,dx\right)$ is equal to:

Let $\displaystyle\int_{-2}^2\left(|\sin x|+[x\sin x]\right)dx=2(3-\cos2)+\beta$, where $[\cdot]$ is the greatest integer function. Then $\beta\sin\left(\dfrac\beta2\right)$ equals:

If $\alpha=1$ and $\beta=1+i\sqrt2$, where $i=\sqrt{-1}$ are two roots of the equation $x^3+ax^2+bx+c=0$, $a,b,c\in\mathbb R$, then $\displaystyle\int_{-1}^1\left(x^3+ax^2+bx+c\right)dx$ is equal to:

The integral $\displaystyle\int_0^1\cot^{-1}\left(1+x+x^2\right)dx$ is equal to:

Let $f$ be a twice differentiable function such that $f(x)=\displaystyle\int_0^x\tan(t-x)\,dt-\int_0^xf(t)\tan t\,dt$, $x\in\left(-\dfrac\pi2,\dfrac\pi2\right)$. Then $f''\left(\dfrac\pi6\right)+12f'\left(-\dfrac\pi6\right)+f\left(\dfrac\pi6\right)$ is equal to ________.

The value of the integral $\displaystyle\int_0^\infty\frac{\log_e(x)}{x^2+4}\,dx$ is:

The value of the integral $\displaystyle\int_{\pi/6}^{\pi/3}\left(\frac{4-\operatorname{cosec}^2x}{\cos^4x}\right)dx$ is:

Let $\left(2^{1-a}+2^{1+a}\right)$, $f(a)$, $\left(3^a+3^{-a}\right)$ be in A.P. and $\alpha$ be the minimum value of $f(a)$. Then the value of the integral $\displaystyle\int_{\log_e(\alpha-1)}^{\log_e(\alpha)}\frac{dx}{\left(e^{2x}-e^{-2x}\right)}$ is :

Let $f:[1,\infty)\to\mathbf R$ be a differentiable function defined as $f(x)=\displaystyle\int_1^xf(t)\,dt+(1-x)(\log_ex-1)+e$. Then the value of $f(f(1))$ is :

The value of the integral $\displaystyle\int_{-\pi/4}^{\pi/4}\left(\frac{32\cos^4x}{1+e^{\sin x}}\right)dx$ is:

Let $A=\begin{bmatrix}1&3&-1\\2&1&\alpha\\0&1&-1\end{bmatrix}$ be a singular matrix. Let $f(x)=\displaystyle\int_0^x\left(t^2+2t+3\right)dt$, $x\in[1,\alpha]$. If M and m are respectively the maximum and the minimum values of $f$ in $[1,\alpha]$, then $3(M-m)$ is equal to :

Let $f:\mathbf R\to\mathbf R$ be such that $f(xy)=f(x)f(y)$, for all $x,y\in\mathbf R$ and $f(0)\ne0$. Let $g:[1,\infty)\to\mathbf R$ be a differentiable function such that
$$x^2g(x)=\int_1^x\left(t^2f(t)-tg(t)\right)dt.$$
Then $g(2)$ is equal to :

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

If $26\left(\dfrac{2^3}3\left({}^{12}C_2\right)+\dfrac{2^5}5\left({}^{12}C_4\right)+\dfrac{2^7}7\left({}^{12}C_6\right)+\dots+\dfrac{2^{13}}{13}\left({}^{12}C_{12}\right)\right)=3^{13}-\alpha$, then $\alpha$ is equal to :

The value of the integral $\displaystyle\int_0^2\frac{\sqrt{x\left(x^2+x+1\right)}}{\left(\sqrt{x+1}\right)\left(\sqrt{x^4+x^2+1}\right)}\,dx$ is equal to :