$\int {{e^x}(1 - \cot x + {{\cot }^2}x)\,dx} $ equals

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

$\int e^{\tan x}(\sec ^{2} x+\sec ^{3} x \sin x) d x$ is equal to

$\int_{\pi/4}^{\pi/2} e^x (\log \sin x + \cot 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 $(GIF)$.

$\int \frac{1+\sin (\log x)}{1+\cos (\log x)} d x=$