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

  • Thread starter FatalFlare
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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?
 

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  • #2
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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.)
 
  • #3
Ray Vickson
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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##).
 

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