The value of $\int \limits_{0}^{\pi}|\cos x|^{3} dx$ is

The value of $\int_{0}^{1} x^{2}(1-x^{2})^{3/2} dx$ is

If $\int_{\sin x}^1 {{t^2}f(t)\;dt = 1 - \sin x} $,$x \in \left( {0,\frac{\pi }{2}} \right)$,then $f\left( {\frac{1}{{\sqrt 3 }}} \right)$ is equal to:

The value of $\mathop {\lim }\limits_{x \to 0} \left( \frac{\int_0^{x^2} \sec^2 t \, dt}{x \sin x} \right)$ is

Given that $\frac{d}{d x}\left[\int_0^{\phi(x)} f(t) d t\right]=f(\phi(x)) \cdot \phi^{\prime}(x)$. If $\int_0^{x^3} f(t) d t = x^2 \sin(2 \pi x)$,then the value of $f(8)$ is

$\int_9^x \frac{f(y)}{y^2} \, dy = 2 \sqrt{x} - 6 \implies f(x) = ?$