Chapter 5.10, Problem 14E

To determine

To find whether the integral ${\displaystyle \underset{1}{\overset{\infty }{\int }}\frac{e^{-\sqrt{x}}}{\sqrt{x}}dx}$ is convergent or divergent.

Answer to Problem 14E

The integral ${\displaystyle \underset{1}{\overset{\infty }{\int }}\frac{e^{-\sqrt{x}}}{\sqrt{x}}dx}$ is convergent.

Explanation of Solution

Given information:

The integral is ${\displaystyle \underset{1}{\overset{\infty }{\int }}\frac{e^{-\sqrt{x}}}{\sqrt{x}}dx}$

Concept used: If the value of the integral is finite, it is convergent otherwise it is divergent

Given, ${\displaystyle \underset{1}{\overset{\infty }{\int }}\frac{e^{-\sqrt{x}}}{\sqrt{x}}dx}$

Put, $\sqrt{x}=y$ , so that $\frac{1}{2\sqrt{x}}dx=dy$

The given integral becomes as:

  $=2{\displaystyle \underset{1}{\overset{\infty }{\int }}{e}^{-y}dy}$

On integration, the integral becomes

  $\begin{array}{l}\text{ =}2{\left[\frac{e^{-y}}{(-1)}\right]}_1^{\infty }=\text{-}2\left[0-e^{-1}\right]\\ =2e^{-1},\text{a}\text{finite}\text{number}\end{array}$

Since the given integral has finite value.

Hence, the given integral is convergent