# Thread: How Do You Derive The Formula of S Sech (x) (dx) = Arcsin (tanh(x))+C?

1. ## How Do You Derive The Formula of S Sech (x) (dx) = Arcsin (tanh(x))+C?

The integrand can be transformed to a square root involving tanh x by use of the Pythagoran property relating sech x and tanh x. Then a very clever trigonometric substitution can be used to rationalize the radical. Confirm that formula works by evaluating the integral on the interval (0,1), then checking by numerical integration.

Can anyone show me how to do this?

2. The pythagorean property is $\text{sech}\, x = \sqrt{1-\tanh^2x}$. I think that the substitution should be $u = \tanh x$. Then $du = \text{sech}^2x\,dx$, and $\int\!\!\text{sech}\,x\,dx = \int\!\!\text{sech}\,x\,\frac{du}{\text{sech}^2x} = \int\!\!\frac1{\text{sech}\,x}\,du =\int\!\!\frac1{\sqrt{1-u^2}}\,du = \arcsin u + C.$