The equation of the normal to the curve $y=x \log x$, which is parallel to the line $2 x-2 y+3=0$, is

$$\int \frac{1+\sin (\log x)}{1+\cos (\log x)} d x=$$

If $\mathrm{w}=\frac{-1+i \sqrt{3}}{2}$, where $\mathrm{i}=\sqrt{-1}$, then the value of $\left(3+w+3 w^2\right)^4$ is

If $|\overline{\mathrm{a}}|=2,|\overline{\mathrm{~b}}|=3$ and $\overline{\mathrm{a}}, \overline{\mathrm{b}}$ are mutually perpendicular vectors, then the area of the triangle whose vertices are $0, a+2 b, a-2 b$ is