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

$\int (x + 3)({x^2} + 6x + 10)^9 \, dx$ equals

If $f(x) = \int \frac{\sin 2x + 2 \cos x}{4 \sin^2 x + 5 \sin x + 1} \, dx$ and $f(0) = 0$, then $f\left(\frac{\pi}{6}\right) =$

The value of the integral $\int \frac{\sin(\ln(2 + 2x))}{x + 1} dx$ is

If $\int \frac{\sqrt{1-x^2}}{x^4} \,dx = A(x)\left(\sqrt{1-x^2}\right)^{m} + c$ for a suitably chosen integer $m$ and a function $A(x)$,where $c$ is a constant of integration,then $(A(x))^{m}$ equals

If $\int \frac{d x}{\sqrt[3]{\sin ^{11} x \cos x}}=-\left(\frac{3}{8} f(x)+\frac{3}{2} g(x)\right)+c$,then: