Indefinite integral using hyperbolic function

Problem:

$\int{\frac{1}{\sqrt{4x^2 + 25}}dx}$

Solution:

The key substitution to use is $x = \frac{5}{2}\sinh y$. Then we have $dx = \frac{5}{2}\cosh y dy$.

$\begin{eqnarray*} & & int{\frac{1}{\sqrt{4x^2 + 25}}dx} \\ &=& \int{\frac{1}{\sqrt{4(\frac{5}{2}\sinh y)^2 + 25}}\frac{5}{2}\cosh y dy} \\ &=& \int{\frac{1}{\sqrt{25\sinh^2 y + 25}}\frac{5}{2}\cosh y dy} \\ &=& \int{\frac{1}{\sqrt{\sinh^2 y + 1}}\frac{1}{2}\cosh y dy} \\ &=& \int{\frac{1}{\cosh y}\frac{1}{2}\cosh y dy} \\ &=& \int{\frac{1}{2} dy} \\ &=& \frac{1}{2}y + C \\ &=& \frac{1}{2}\sinh^{-1}\frac{2x}{5} + C \\ \end{eqnarray*}$