Integration of sqrt(x^2-9)/x

[FatalFlare]

I used trig substitution and got sqrt(x^2-9)+3*arcsin(3/x) which seems to be incorrect when I check it in my calculator and the textbook. I made a right triangle where one of the legs was sqrt(x^2-9) and it so happens that if you switch the leg the answer becomes sqrt(x^2-9) - 3*arctan(sqrt(x^2-9)/3)) which is the correct answer. It shouldn't matter which leg I choose but it does why?

 

[Mark44]

I moved this thread, which was originally posted in the HW section. It is not so much a homework problem as a question about why an answer can appear in two different forms. (Also, it might have drawn a warning in the HW section, as it was not posted using the homework template.)

 

[Ray Vickson]

I used trig substitution and got sqrt(x^2-9)+3*arcsin(3/x) which seems to be incorrect when I check it in my calculator and the textbook. I made a right triangle where one of the legs was sqrt(x^2-9) and it so happens that if you switch the leg the answer becomes sqrt(x^2-9) - 3*arctan(sqrt(x^2-9)/3)) which is the correct answer. It shouldn't matter which leg I choose but it does why?

Both $\sqrt{x^2-9}+3\arcsin(3/x)$ and $\sqrt{x^2-9} - 3\arctan(\sqrt{x^2-9}/3))$ have the same derivative, namely, your original integrand (assuming $x>3$).