$\int \frac{3x^2}{x^6 + 1} dx = $

$\int \frac{\sin 2x}{a^2 + b^2 \sin^2 x} \, dx = $

Let $f(x) = \int \frac{dx}{x^{2/3} + 2x^{1/2}}$ be such that $f(0) = -26 + 24 \log_{e}(2)$. If $f(1) = a + b \log_{e}(3)$, where $a, b \in \mathbb{Z}$, then $a + b$ is equal to:

If $\int \frac{x^3}{\sqrt{1+x^2}} d x=A(1+x^2)^{\frac{3}{2}}+B(1+x^2)^{\frac{1}{2}}+C$,then $A+B=$

$\int \frac{x^{9/2}}{\sqrt{1+x^{11}}} dx$ is equal to -

$\int \sqrt{\operatorname{cosec} x - \sin x} \, dx$ is equal to