Julia drove at a constant speed and went 25 miles in ½ an hour. What was her speed in miles per hour?

Solve the following equations:
${\log }_{2}x+8=13$

Evaluate the indefinite integral. (Remember to use absolute values where appropriate. Use C for the constant of integration.)
${\displaystyle \int \frac{dx}{qx+s}(q\ne 0)}$

$\dot{x}=x-y-x(x^2+y^2)$
$\dot{y}=y+x-y(x^2+y^2)$
1. Determine the equilibrium points
2. Show that this system has a periodic solution. Use the following substitution $x=r\cos (\varphi ),y=r\sin (\varphi )$.
3. Give the explicit solution for this system. Also determine the period of this solution.

Solve equation:
${\displaystyle 2x+1-12=0}$

Is ${\displaystyle \sin (\omega t+\pi )=-\sin \omega t}$ always true no matter what ${\displaystyle \omega }$ is?
For example, it's true that ${\displaystyle \sin (t+\pi )=-\sin \left(t\right)}$. You flip one over the x-axis, and you get the other. But ${\displaystyle \sin (2t+\pi )=\sin \left(2t\right)}$. Because ${\displaystyle \sin \left(2t\right)}$ has a period of ${\displaystyle \pi }$, so if you translate it by π, you are still gonna get ${\displaystyle \sin \left(2t\right)}$

How do you simplify ${\displaystyle (\frac{1}{2}+\frac{2}{5i})+(\frac{1}{20}-\frac{1}{5i})}$?

Is $n^{\log c}=c^{\log n}$ true?
If so, please explain.