Let $f: R \rightarrow R$ be a function defined by $f(x)=\frac{x}{(1+x^4)^{1/4}}$ and $g(x)=f(f(f(f(x))))$. Then find the value of $18 \int_0^{\sqrt{2\sqrt{5}}} x^3 g(x) dx$.

$\int_{1}^{e} \frac{dx}{x(1+\log x)^{2}} =$

$\int_1^{e^2} \frac{dx}{x(1+\log x)^2} = $

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:

Evaluate the following integral:
$\int_{4}^{9} \frac{\sqrt{x}}{\left(30-x^{\frac{3}{2}}\right)^{2}} d x$

It is given that $\frac{d}{dt}(t \log t - t) = \log t$. Then, $\exp \left( \int_0^1 2x \log(1+x^2) dx \right) = $