Evaluate the integral. \int \frac{2x^{2}+7x-3}{x-2}dx

Find all rational zeros of the polynomial, and write the polynomial in factored form.
${\displaystyle P\left(x\right)=4x^4-37x^2+9}$

Let R be the rectangle consisting of all points (x, y), such that ${\displaystyle 0\le x\le 4,1\le y\le 3}$, and calculate the double integral.
${\displaystyle {\int }_R\int (2x+3y)dxdy}$

The following integrals require a preliminary step such as a change of variables before using the method of partial fractions. Evaluate these integrals.
${\displaystyle \int \frac{dx}{1+e^x}}$

Evaluate the following integrals or stat that they diverge.
${\displaystyle {\int }_{-\mathrm{\infty }}^{-1}\frac{dx}{x^3}}$

Evaluate the integral.
${\displaystyle \int \frac{3^{2x+1}}{2^{x+3}}dx}$

Determine if a linear transformation function is
$T:M_{33},$
$T\left(A\right)=\left[\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right]A$

We need to verify theorem 5 Focus-Directrix definition: ${\displaystyle 0<e<1\text{and}c=d(e-2-1)}$