On the interval $[0, 1]$, the function $f(x) = x^{25}(1 - x)^{75}$ attains its maximum value at the point $x = $....

If $f(x) = \int \dfrac{x}{(x^2 + 4)(x^2 + 9)}\, dx$ and $f(0) = \dfrac{1}{5}\log\left(\dfrac{2}{3}\right)$, then $f(1) = $

$\int \dfrac{x^4(x^{10} - 1)}{x^{20} + 3x^{10} + 1}\, dx = \cdots$

$\int \left(e^{\log(\sin x)} + \cos x\right) x\, dx = $