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Can you integrate $\tanh(x)$ by parts?

I realize it's probably easier to use a derivative substitution, but can it be done?

Essentially I just want some clarification that I'm using integration by parts correctly.

$u=\tanh(x)$, $u'=sech^2(x)$

$v=x$, $v'=1$

$$\int uv' = uv - \int u'v$$ $$\int \tanh(x)\,dx = x\,\tanh(x) - \int x\, sech ^2(x)\,dx$$

Idris Addou
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lost
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  • It seems that you want to make life very compicated ! Cheers. – Claude Leibovici Oct 25 '16 at 07:01
  • Roughly speaking, one can apply integration by parts to anything in any way one wants, by if that results in the new integral being even worse than before, then it was time and effort wasted. So you effectively asked two questions here. (1) Can you apply the by parts formula to this integral? -- Yes. (2) Can you find the answer to this integral problem using the parts method? -- I'd say no. At least not the way you started. And I can't actually think of any other reasonable way of doing so. – zipirovich Oct 25 '16 at 12:29

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Why not simply proceed as follows? \begin{align*} \int \tanh(x) dx &=\int \frac{e^x+e^{-x}}{e^x-e^{-x}} dx\\ &=\int \frac{d(e^x-e^{-x})}{e^x-e^{-x}}\\ &=\ln\, \left|\,e^x-e^{-x}\right| + C\\ \end{align*}

Gordon
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