Determine whether the given set S is a

${\int }_1^3\frac{8x}{(2x+4)*3}dx$

In a jet engine a diffuser decreases the kinetic energy of the air entering the engine.  The air enters a the diffuser at 132 kPa and 25°C with a velocity of 502 m/s and leaves at 229 kPa and 64°C.  The heat lost from the diffuser is estimated to be 6 kJ/kg.  Determine the exit velocity in m/s to 1 decimal place.  Take the specific heat at constant pressure of air to be 1.009 kJ/(kgK).

Prove that ${\displaystyle n\mid \varphi (a^n-1)}$ in "Topics in Algebra 2nd Edition" by I. N. Herstein. Any natural solution that uses $Aut\left(G\right)$

Simple Linear Regression - Difference between predicting and estimating?
Here is what my notes say about estimation and prediction:
Estimating the conditional mean
"We need to estimate the conditional mean ${\displaystyle {\beta }_0+{\beta }_1{x}_0}$ at a value ${\displaystyle x_0}$, so we use ${\displaystyle \widehat{Y_0}=\widehat{{\beta }_0}+\widehat{{\beta }_1}{x}_0}$ as a natural estimator." here we get
$\widehat{Y_0}~N({\beta }_0+{\beta }_1{x}_0,{\sigma }^2{h}_{00}) \text{where }{h}_{00}=\frac{1}{n}+\frac{{(x_0-\bar{x})}^2}{(n-1)s_x^2}$
with a confidence interval for ${\displaystyle E\left(Y_0\right)={\beta }_0+{\beta }_1{x}_0}$ is
${\displaystyle (\widehat{b_0}+\widehat{b_1}{x}_0-cs\sqrt{h_{00}},\widehat{b_0}+\widehat{b_1}{x}_0+cs\sqrt{h_{00}})}$
where ${\displaystyle c=t_{n-2,1-\frac{\alpha }{2}}}$ Where these results are found by looking at the shape of the distribution and at ${\displaystyle E\left(\widehat{Y_0}\right)}$ and ${\displaystyle var\left(\widehat{Y_0}\right)}$
Predicting observations
"We want to predict the observation ${\displaystyle Y_0={\beta }_0+{\beta }_1{x}_0+{ϵ}_0}$ at a value ${\displaystyle x_0}$"
$E(\widehat{Y_0}-Y_0)=0 \text{and }var(\widehat{Y_0}-Y_0)={\sigma }^2(1+h_{00})$
Hence a prediction interval is of the form
${\displaystyle (\widehat{b_0}+\widehat{b_1}{x}_0-cs\sqrt{h_{00}+1},\widehat{b_0}+\widehat{b_1}{x}_0+cs\sqrt{h_{00}+1})}$

I want to know how to find the vertices of the conic equation 

20x^2 +4y^2-800=0

Prove if $F\left(\sqrt[n]{a}\right)$ is unramified or totally ramified in certain conditions