$$ \int \frac{\mathrm{e}^{\tan ^{-1} 2 x}}{1+4 x^2}= $$

If the equation of the median through vertex $\mathrm{A}(3, \mathrm{k})$ of $\triangle \mathrm{ABC}$ with vertices $\mathrm{B}(2,1)$ and $\mathrm{C}(-4,5)$ is $x+4 y=\mathrm{p}$, then $\mathrm{k}=$ where p and k are constants

If $\overline{\mathrm{a}}$ and $\overline{\mathrm{b}}$ are unit vectors and $\theta$ is the angle between them, then $\tan \frac{\theta}{2}=$

A manufacturer produces $x$ items per week at a total cost of ₹ $\left(x^2+78 x+2500\right)$. The price per unit is given by $8 x=600-\mathrm{p}$ where ' p ' is the price of each unit. Then the maximum profit obtained is