$\int {{e^x}\sin x(\sin x + 2\cos x)} \,dx = $

$\int {{e^{{x^2}}}} \cdot {e^x}\left( {2{x^2} + x + 1} \right)dx = {e^{{x^2} + x}}\left( {f\left( x \right)} \right) + c$ where $c$ is the constant of integration. If the minimum value of $f(x)$ is equal to $m$,then find the value of $\left[ { - \frac{1}{m}} \right]$,where $[\cdot]$ denotes the Greatest Integer Function $(GIF)$.

Find : $\int \frac{(x^{2}+1) e^{x}}{(x+1)^{2}} d x$

$\int e^{2x} \frac{(\sin 2x \cos 2x - 1)}{\sin^2 2x} \, dx =$

The value of the integral $\int_{1}^{2} e^{x}\left(\log _{e} x+\frac{x+1}{x}\right) d x$ is

$\int e^x \left(\frac{x+2}{x+4}\right)^2 dx =$