Integrating hyperbolic functions

[peripatein]

Hi,
I am trying to integrate (tanh(x)+coth(x))/((cosh(x))^2)
I am substituting u=tanh(x), du=dx/((cosh(x))^2)
and end up with 1/2(tanh(x))^2 + ln |tanh(x)| + C
which is incorrect. What am I doing wrong??

 

[peripatein]

Does someone have an idea what is stymying my answer?

 

[SteamKing]

Why did you chose u = tanh(x)? What happens if you expand (tanh(x) + coth (x))?

 

[peripatein]

I used u=tanhx, as 1/(coshx)^2 is its derivative.

 

[Dick]

Does someone have an idea what is stymying my answer?

There is nothing wrong with your answer. If the book is giving one that looks different it may differ from yours by a constant.

 

[peripatein]

Online calculators claim the integral to be -1/2*(coshx)^2 + ln |tanhx| + c.
1/2*(tanhx)^2 (which is the first term in my answer) is not equal to -1/2*(coshx)^2, is it?

 

[Dick]

Online calculators claim the integral to be -1/2*(coshx)^2 + ln |tanhx| + c.
1/2*(tanhx)^2 (which is the first term in my answer) is not equal to -1/2*(coshx)^2, is it?

Wolfram Alpha gives the first term as -(sechx)^2/2 and that does differ from (tanhx)^2/2 by a constant. What's the constant? Are you sure the online calculator isn't saying -1/(2*(coshx)^2)? You should use more parentheses when you write something like -1/2*(coshx)^2. It's ambiguous.

 

[peripatein]

I am not following your argument. Is the answer which Wolfram's calculator yields equal to mine?
My answer is: (0.5)(tanh(x))^2 + ln |tanh(x)| + C
Wolfram's calculator's answer: (-0.5)(sech(x)^2) + ln [tanh(x)] + C

 

[Dick]

I am not following your argument. Is the answer which Wolfram's calculator yields equal to mine?
My answer is: (0.5)(tanh(x))^2 + ln |tanh(x)| + C
Wolfram's calculator's answer: (-0.5)(sech(x)^2) + ln [tanh(x)] + C

They are only 'equal' if you consider the '+C' part. (tanhx)^2+(sechx)^2=1. Use that identity.

 

[peripatein]

I see. Thanks a lot!