$\int_1^{e} \frac{e^x}{x}(1+x \log x) d x=$

If $\int {{e^{{x^2}}}\left( {2 - \frac{1}{{{x^2}}}} \right)dx = {e^{{x^2}}}f(x) + C} $ and $f\left( {\frac{1}{2}} \right) = 2$,then $f(1)$ is equal to (where $C$ is an arbitrary constant).

$\int {\frac{{(x + 3){e^x}}}{{{{(x + 4)}^2}}}\,dx} = \,$

If $\int e^{2x} \frac{2(\sin 2x \cos 2x - 1)}{2 \sin^2 2x} dx = A e^{2x} \cot 2x + c$ (where $c$ is the constant of integration), then $A^3 =$ ?

$\int e^{x \operatorname{cosec} x} \cdot \operatorname{cosec} x \cdot(1-x \cot x) \, dx =$

The value of $\int \frac{e^{\tan ^{-1}(x)}}{1+x^2} \left[\left(\sec ^{-1} \sqrt{1+x^2}\right)^2+\cos ^{-1}\left(\frac{1-x^2}{1+x^2}\right)\right] d x$,for $x>0$ is