$$\int \frac{1}{\cos ^3 x \sqrt{\sin 2 x}} d x=$$

The shortest distance (in units) between the lines $$\frac{x+1}{3}=\frac{y+2}{1}=\frac{z+1}{2}$$ and $$\bar{r}=(2 \hat{i}-2 \hat{j}+3 \hat{k})+\lambda(\hat{i}+2 \hat{j})$$ is

The maximum value of $$z=7 x+8 y$$ subject to the constraints $$x+y \leq 20, y \geq 5, x \leq 10, x \geq 0, y \geq 0$$ is

The value of $$\int_\limits0^\pi\left|\sin x-\frac{2 x}{\pi}\right| \mathrm{d} x$$ is