What is the integral of sinh^2(x) dx?

To find the integral of ( \sinh^2(x) , dx ), it's useful to apply a hyperbolic identity to simplify the integration process. The identity states:

[

\sinh^2(x) = \frac{1}{2}(\cosh(2x) - 1)

]

This allows us to rewrite the integral as:

[

\int \sinh^2(x) , dx = \int \frac{1}{2} (\cosh(2x) - 1) , dx

]

Breaking this down further, we integrate each term separately:

1. The integral of ( \cosh(2x) ) is ( \frac{1}{2} \sinh(2x) ).

2. The integral of ( 1 ) is simply ( x ).

Putting these results together, we have:

[

\int \sinh^2(x) , dx = \frac{1}{2} \left( \frac{1}{2} \sinh(2x) - x \right) + C = \frac{1}{2} \sinh(2x) - \frac{1