If $F(x) = \int_{x^2}^{x^3} \log t \, dt$ $(x > 0)$,then $F'(x) = $

$\lim _{x \rightarrow 0} \frac{48}{x^4} \int _{0}^{x} \frac{t^3}{t^6+1} dt$ is equal to $.......$.

If $\int_0^{t^2} xf(x)dx = \frac{2}{5}t^5, t > 0,$ then $f\left( \frac{4}{25} \right) = $

Let the slope of the line $45 x+5 y+3=0$ be $27 r_1+\frac{9 r_2}{2}$ for some $r_1, r_2 \in R$. Then $\lim_{x \rightarrow 3} \left( \int_3^x \frac{8 t^2}{\frac{3 r_2 x}{2}-r_2 x^2-r_1 x^3-3 x} dt \right)$ is equal to ...................

If $\int_0^{2a} x^2 \sqrt{2ax-x^2} dx = ka^4$, then $k : \pi =$ (in $:8$)

$\int_0^{\pi/6} \cos^4 3\theta \cdot \sin^2 6\theta \, d\theta$ is equal to