$\int_{1}^{2} \frac{\cos(\log x)}{x} \, dx = $

$\int_0^1 \frac{\tan^{-1} x}{1 + x^2} \,dx = $

$\int_{\pi /3}^{\pi /2} \frac{\sqrt{1 + \cos x}}{(1 - \cos x)^{5/2}} \,dx = $

Let $y=f(x)$ be a thrice differentiable function in $(-5,5)$. Let the tangents to the curve $y=f(x)$ at $(1, f(1))$ and $(3, f(3))$ make angles $\frac{\pi}{6}$ and $\frac{\pi}{4}$,respectively with the positive $x$-axis. If $27 \int_1^3\left(\left(f^{\prime}(t)\right)^2+1\right) f^{\prime \prime}(t) d t=\alpha+\beta \sqrt{3}$,where $\alpha$ and $\beta$ are integers,then the value of $\alpha+\beta$ equals:

$\int_0^{\pi /6} \frac{\sin x}{\cos^3 x} \, dx = $

$\int_0^{\frac{\pi}{4}} \frac{\sec ^2 x}{(1+\tan x)(2+\tan x)} d x=$