Chapter 9, Problem 37RE

To determine

To evaluate : The improper integral or state that it diverges.

Answer to Problem 37RE

The answer is 2.

Explanation of Solution

Given information :

The given improper integral is ${\displaystyle {\int }_1^{\infty }\frac{dx}{x^{3/2}}}$ .

Calculation :

Since, if $f\left(x\right)$ is continuous on $\left[a,\infty \right)$ . Then,

  ${\displaystyle {\int }_a^{\infty }f\left(x\right)}dx=\underset{b\to b}{\lim }{\displaystyle {\int }_a^{b}f\left(x\right)}dx$ .

Therefore,

  $\begin{array}{l}{\displaystyle {\int }_1^{\infty }\frac{dx}{x^{3/2}}}=\underset{t\to \infty }{\lim }{\displaystyle {\int }_1^t{x}^{-3/2}}dx\\ {\displaystyle {\int }_1^{\infty }\frac{dx}{x^{3/2}}}=\underset{t\to \infty }{\lim }{\left[-2x^{-1/2}\right]}_1^t\\ {\displaystyle {\int }_1^{\infty }\frac{dx}{x^{3/2}}}=\underset{t\to \infty }{\lim }\left[\frac{-2}{\sqrt{t}}+\frac{1}{1}\right]\\ {\displaystyle {\int }_1^{\infty }\frac{dx}{x^{3/2}}}=\left[0+2\right]\\ {\displaystyle {\int }_1^{\infty }\frac{dx}{x^{3/2}}}=2\end{array}$

Hence,

The answer is 2.