Integral of sech x: Typo Error Check

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SUMMARY

The discussion centers on the integral of sech x and a potential typo in the online notes regarding the final expression. The correct formulation is confirmed to be 2tan^(-1)(tanh(x/2)) + C, following the substitution u = (x/2) and s = tanh(u). The user also clarifies that the differential du corresponds to 0.5sech^2(x/2)dx, which necessitates pulling a factor of 2 out of the integral. This confirms the accuracy of the original statement in the notes.

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DryRun
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Homework Statement
I believe there might be a mistake in this statement from my online notes. A typo error in the last line, maybe?
http://s1.ipicture.ru/uploads/20111209/K6L7AWhS.jpgThe attempt at a solution
In the last line, from what i worked out, it should be (1/2)tan^(-1)(tanh... instead of 2tan^(-1)(tanh... Correct?
 
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Nope it is right. I would make two substitutions instead of one just because it is easier for me to think through that way. I would substitute u=(x/2) and then s=tanh(u) yielding the simple integral 2∫1/(s^2+1)ds=2tan^-1(s)=2tan^-1(tanh(u))=2tan^-1(tanh(x/2))+C

edit: The du for your equation would be .5sech^2(x/2)dx so you would pull a 2 out in front of the integral.
 
Thank you, PCSL. I see it now.
 

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