$\int_1^e \frac{1 + \log x}{x} \, dx = $

The value of the integral $\int_0^{\pi /4} \frac{\sqrt{\tan x}}{\sin x \cos x} \, dx$ equals

Evaluate the definite integral $\int_{0}^{1} x e^{x^{2}} d x$.

$\int_1^2 x \sqrt{4-x^2} \, dx =$

$\int_0^1 \frac{dx}{(3x+2)+\sqrt{3x+2}} = $ . . . . . . .

If $\alpha = \int_0^1 \left(e^{9x + 3 \tan^{-1} x}\right) \left(\frac{12 + 9x^2}{1 + x^2}\right) dx$,where $\tan^{-1} x$ takes only principal values,then the value of $\left(\log_e |1 + \alpha| - \frac{3\pi}{4}\right)$ is