Let for $f(x)=7{\tan }^{8}x+7{\tan }^{6}x-3{\tan }^{4}x-3{\tan }^{2}x,{\mathrm{I}}_1={\int }_0^{\pi /4}f(x)\mathrm{d}x$ and ${\mathrm{I}}_2={\int }_0^{\pi /4}xf(x)\mathrm{d}x$. Then $7{\mathrm{I}}_1+12{\mathrm{I}}_2$ is equal to :

Let $\mathrm{f}:\mathrm{R}\to \mathrm{R}$ be a twice differentiable function such that $f(2)=1$. If $\mathrm{F}(\mathrm{x})=\mathrm{x}f(\mathrm{x})$ for all $\mathrm{x}\in \mathrm{R}$, $\underset{0}{\overset{2}{\int }}x{F}^{\prime }(x)dx=6$ and $\underset{0}{\overset{2}{\int }}{x}^2{F}^{\prime \prime }(x)dx=40$, then $F^{\prime }(2)+\underset{0}{\overset{2}{\int }}F(x)dx$ is equal to :

Let $f$ be a real valued continuous function defined on the positive real axis such that $g(x)=\underset{0}{\overset{x}{\int }}tf(t)dt$. If $g\left(x^3\right)=x^6+x^7$, then value of $\sum _{r=1}^{15}f\left(r^3\right)$ is :

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

If $\mathrm{I}={\int }_0^{\frac{\pi }{2}}\frac{{\sin }^{\frac{3}{2}}x}{{\sin }^{\frac{3}{2}}x+{\cos }^{\frac{3}{2}}x}\mathrm{d}x$, then ${\int }_0^{2I}\frac{x\sin x\cos x}{{\sin }^{4}x+{\cos }^{4}x}\mathrm{d}x$ equals :

If $I(m,n)={\int }_0^1{x}^{m-1}(1-x{)}^{n-1}dx,m,n>0$, then $I(9,14)+I(10,13)$ is

If ${\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}\frac{96x^2{\cos }^{2}x}{\left(1+e^x\right)}\mathrm{d}x=\pi \left(\alpha {\pi }^2+\beta \right),\alpha ,\beta \in \mathbb{Z}$, then $(\alpha +\beta {)}^2$ equals

The integral $\underset{-1}{\overset{\frac{3}{2}}{\int }}\left(|{\pi }^{2}x\sin (\pi x)\right|)dx$ is equal to:

Let f(x) be a positive function and $I_1=\underset{-\frac{1}{2}}{\overset{1}{\int }}2xf(2x(1-2x))dx$ and $I_2=\underset{-1}{\overset{2}{\int }}f(x(1-x))dx$. Then the value of $\frac{I_2}{I_1}$ is equal to ________

The integral ${\int }_0^{\pi }\frac{(x+3)\sin x}{1+3{\cos }^{2}x}dx$ is equal to

Let $f:[1,\infty )\to [2,\infty )$ be a differentiable function. If $10{\int }_1^{1}f(\mathrm{t})\mathrm{dt}=5xf(x)-x^5-9$ for all $x⩾1$, then the value of $f(3)$ is :

Let $(a, b)$ be the point of intersection of the curve $x^2=2y$ and the straight line $y-2 x-6=0$ in the second quadrant. Then the integral $\mathrm{I}={\int }_{\mathrm{a}}^{\mathrm{b}}\frac{9x^2}{1+5^x}\mathrm{d}x$ is equal to :

$4{\int }_0^1\left(\frac{1}{\sqrt{3+x^2}+\sqrt{1+x^2}}\right)dx-3{\log }_e(\sqrt{3})$ is equal to :

Let the domain of the function $f(x)={\log }_2{\log }_4{\log }_6\left(3+4x-x^2\right)$ be $(a, b)$. If ${\int }_0^{b-a}\left[x^2\right]dx=p-\sqrt{q}-\sqrt{r},p,q,r\in \mathbb{N},\gcd (p,q,r)=1$, where $[\cdot ]$ is the greatest integer function, then $p+q+r$ is equal to

Let $f(x)+2f\left(\frac{1}{x}\right)=x^2+5$ and $2g(x)-3g\left(\frac{1}{2}\right)=x,x>0$. If $\alpha ={\int }_1^{2}f(x)\mathrm{d}x$, and $\beta ={\int }_1^{2}g(x)\mathrm{d}x$, then the value of $9\alpha +\beta$ is :

The value of \int_\limits{-1}^1 \frac{(1+\sqrt{|x|-x}) e^x+(\sqrt{|x|-x}) e^{-x}}{e^x+e^{-x}} d x is equal to

The integral ${\int }_0^{\pi }\frac{8xdx}{4{\cos }^{2}x+{\sin }^{2}x}$ is equal to

Let $|z_1-8-2i|\le 1$ and $|z_2-2+6i|\le 2$, $z_1,z_2\in \mathbb{C}$. Then the minimum value of $|z_1-z_2|$ is :

Let $z_1,z_2$ and $z_3$ be three complex numbers on the circle $|z|=1$ with $\arg \left(z_1\right)=\frac{-\pi }{4},\arg \left(z_2\right)=0$ and $\arg \left(z_3\right)=\frac{\pi }{4}$. If ${\left|z_1{\bar{z}}_2+z_2{\bar{z}}_3+z_3{\bar{z}}_1\right|}^2=\alpha +\beta \sqrt{2},\alpha ,\beta \in Z$, then the value of ${\alpha }^2+{\beta }^2$ is :

Let the curve $z(1+i)+\bar{z}(1-i)=4,z\in C$, divide the region $|z-3|\le 1$ into two parts of areas $\alpha$ and $\beta$. Then $|\alpha -\beta |$ equals :