Sue Pemberton Solutions for Chapter: Integration, Exercise 8: EXERCISE 9H

Show that the following improper integral has a finite value and find the value.

${\int }_1^{\infty }\left(\frac{2}{x^2}+\frac{4}{(x+2{)}^3}\right)dx$

The diagram shows part of the curve $y=\frac{20}{(2x+5{)}^2}$.

Show that as $p\to \infty$, the shaded area tends to the value $2$.

Show that the following improper integral does not exist.

${\int }_4^{\infty }\frac{6}{\sqrt{x}} dx$ 

Show that the following improper integral does not exist.

${\int }_0^{\infty }\frac{4}{x\sqrt{x}} dx$ 

Show that the following improper integral does not exist.

${\int }_0^9\frac{12}{x^2\sqrt{x}} dx$ 

Show that the following improper integral does not exist.

${\int }_5^{\infty }\frac{2}{\sqrt{x+4}} dx$ 

Show that the following improper integral does not exist.

${\int }_{\frac{1}{2}}^2\frac{5}{(2x-1{)}^2} dx$ 

Show that the following improper integral does not exist.

${\int }_0^{25}\left(\sqrt{x}+\frac{1}{x^2}\right)\mathrm{d}x$