Chapter A6, Problem 58E

To determine

To Evaluate: the integrals.

Answer to Problem 58E

The result is $\mathrm{0.39.227}$

Explanation of Solution

Given:

  ${\displaystyle \underset{1}{\overset{4}{\int }}\frac{8\cosh \sqrt{x}}{\sqrt{x}}}dx$

Calculation:

assume a variable $u$ such that

  $u=\sqrt{x},du=\frac{dx}{2\sqrt{x}}$

The limit now changes as follows:

Lower Limit $\sqrt{1}=1$

Upper Limit $\sqrt{4}=2$

Therefore the integral can now be evaluated as,

  $\begin{array}{l}{\displaystyle \underset{1}{\overset{4}{\int }}\frac{8\cosh \sqrt{x}}{\sqrt{x}}}.dx=16{\displaystyle \underset{1}{\overset{2}{\int }}\cosh u.du}\\ =16{\left[\mathrm{sinhu}\right]}_1^2\\ =16\left(\left(\sinh 2\right)-\left(\sinh 1\right)\right)\\ =16\left[\left(\frac{e^2-e^{-2}}{2}\right)-\left(\frac{e^1-e^{-1}}{2}\right)\right]\\ =8\left(e^2-e^{-2}-e+e^{-1}\right)\\ \approx \mathrm{0.39.227}\end{array}$

Conclusion:

The result is $\mathrm{0.39.227}$