Find the average value h_{ave} of the function h on the given interval. NS

Evaluate the line integral, where C is the given curve
C xy ds
C: $x=t^2,y=2t,0\le t\le 5$

$(x^2+2xy-4y^2)dx-(x^2-8xy-4y^2)dy=0$

Determine the following indefinite integral.
${\displaystyle \int (1+3\cos 0)d0}$

To calculate: The simplified form of the expression , ${\displaystyle \sqrt{\frac{8.0\times {10}^{12}}{2.0\times {10}^4}}}$

Let $f:[a,b]\to \mathbb{R}$ be monotone increasing, $F:[a,b]\to \mathbb{R}$ such that $F(x):={\int }_{[a,x]}f$ and $x_0\in [a,b]$. Then $F$ is differentiable at $x_0$ if and only if $f$ is continuous at $x_0$

What are the equations of rotated and shifted ellipse, parabola and hyperbola in the general conic sections form?
How will look the general conic sections equation ${\displaystyle A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0}$ in case of rotated and shifted from coordinates origin ellipse, parabola and hyperbola?
I need a formulas for coefficients A, B, C, D, E and F for ellipse, hyperbola and parabola. I did it for not-rotated conic sections at the origin of coordinates but have a difficults with rotated and shifted.
For i.e. standart ellipse equation ${\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=1}$ gives me general equation ${\displaystyle \frac{1}{a^2}{x}^2+0xy+\frac{1}{b^2}{y}^2+0x+0y-1=0}$
I need the same in case if ellipse located in position (h;k) and rotated on some angle ${\displaystyle \alpha }$ from positive X axis. And the same for parabola and hyperbola.

To solve:
${\displaystyle 3(x-1.1)=5x-5.3}$