Can somebody tell me if this is right?

I must solve:

$\displaystyle \displaystyle\int_{}^{}\displaystyle\frac{dx}{\sqr t[ ]{9+x^2}}$

And I used the substitutions:

$\displaystyle x=3sinht$

$\displaystyle dx=3coshtdt$

And the identity:

$\displaystyle cosh^2t-sinh^2t=1\Rightarrow{\cosht=\sqrt[ ]{1+sinh^2t}}$

$\displaystyle \displaystyle\int_{}^{}\displaystyle\frac{dx}{\sqr t[ ]{9+x^2}}=\displaystyle\int_{}^{}\displaystyle\frac {3coshtdt}{\sqrt[ ]{3^2+(3sinht)^2}}=\displaystyle\int_{}^{}\displays tyle\frac{3coshtdt}{\sqrt[ ]{3^2(1+sinh^2t)}}=\displaystyle\int_{}^{}\displays tyle\frac{coshtdt}{cosht}=\displaystyle\int_{}^{}d t=t+C=argsh(\displaystyle\frac{x}{3})+C$

Bye there, and thanks for posting.