Chapter A6, Problem 60E

To determine

To Evaluate: the integrals.

Answer to Problem 60E

The result is $5.295$

Explanation of Solution

Given:

  ${\displaystyle \underset{0}{\overset{\ln 10}{\int }}4{\sinh }^2\left(\frac{x}{2}\right)}dx$

Calculation:

Therefore the integral can now be evaluated as,

  $\begin{array}{l}{\displaystyle \underset{0}{\overset{\ln 10}{\int }}4{\sinh }^2\left(\frac{x}{2}\right)}.dx={\displaystyle \underset{0}{\overset{\ln 10}{\int }}4\left(\frac{\cosh x-1}{2}\right)}.dx\\ =2{\displaystyle \underset{0}{\overset{\ln 10}{\int }}\left(\cosh x-1\right)}dx\\ =2{\left[\sinh x-x\right]}_0^{\ln 10}\\ =2\left[\left(\left(\sinh \left(\ln 10\right)-\ln \left(10\right)\right)-\left(\sinh \left(0\right)-0\right)\right)\right]\\ =\left(e^{\ln 10}-e^{-ln10}-2\ln \left(10\right)\right)\\ =9.9+2\ln 10\\ \approx 5.295\end{array}$

Conclusion:

The result is $5.295$