Integrating by substituting hyperbolic functions

Question

I am trying to integrate:

$$ \int\sqrt{x^2-9}dx $$

by substituting in: $$ x=\sinh{t} $$

I know you can solve it by using $$ x=\sec{t} $$

But I want to solve it using the identity: $$ \sinh^2{x}=\frac{1}{2}(\cosh{2x}-1) $$

Could you help?

Answer 1

Let $x=\frac{3(e^t+e^{-t})}{2}$, where $t\geq0$.

Thus, $$\int\sqrt{x^2-9}dx=\int\sqrt{\frac{9(e^{2t}+e^{-2t}+2)}{4}-9}\cdot\frac{3(e^t-e^{-t})}{2}dt=$$ $$=\frac{9}{4}\int|e^t-e^{-t}|(e^t-e^{-t})dt=\frac{9}{4}\int(e^t-e^{-t})^2dt.$$

Can you end it now?