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

$\int {{e^{2x}}\frac{{1 + \sin 2x}}{{1 + \cos 2x}}} \,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 $(GIF)$.

If $I = \int e^{\sin \theta} (\log \sin \theta + \operatorname{cosec}^2 \theta) \cos \theta \, d\theta$,then $I$ is equal to

Integrate the function: $\frac{(x-3) e^{x}}{(x-1)^{3}}$

The solution of $\frac{dy}{dx} = e^x(\sin x + \cos x)$ is