What is an improper integral?

Evaluate the improper integral ∫ from 1 to ∞ of 1/x^2 dx.

Evaluate the following integral or state that it diverges:

${\int }_5^{\infty }\frac{1}{t\ln t}dt$

Let the gamma function be defined by $\Gamma \left(p\right)={\int }_0^{\infty }{x}^{p-1}{e}^{-x}dx$. Evaluate $\Gamma \left(\frac{3}{2}\right)$ by using the substitution $x=u^2$. (Hint: you may use the Gaussian integral identity ${\int }_0^{\infty }{e}^{-u^2}du=\frac{\sqrt{\pi }}{2}$.)

Complete the square to evaluate ${\int }_{-\infty }^{\infty }{e}^{-(2x^2-3x+1)}dx$ given the Gaussian integral formula ${\int }_{-\infty }^{\infty }{e}^{-a{x}^2}dx=\sqrt{\frac{\pi }{a}}$.