$\frac{1}{2} \int_2^3 \frac{2 x}{x^2+1} d x=$ . . . . . . .

$\int_0^{\frac{\pi}{4}} \frac{\cos^2 x \sin^2 x}{(\cos^3 x + \sin^3 x)^2} \, dx =$

The value of $\int_1^{e^2} \frac{dx}{x(1 + \ln x)^2}$ is

$\int_0^1 \frac{x^7}{\sqrt{1 - x^4}} dx$ is equal to

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\limits_0^1 {\frac{{{{\tan }^{ - 1}}x}}{x}\,dx} = $