Integrating hyperbolic tanh x - tanh^3 x

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Help_Me_Alex
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I have just shown by writing in terms of sinh x and cosh x that

tanh^2 x = 1 - sech^2 x and

d/dx (tanh x) = sech^2 x

The question is to integrate (tanh x - tanh^3 (x)) which I could use a little help with.

I have got this into the form
sinh x / cosh^3 x or tanhx sech^2 x and tried integrating by parts but I got minus the exact same integral. If someone could just point out where I'm going wrong.

Thanks
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TeeEm
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(Original post by Help_Me_Alex)
I have just shown by writing in terms of sinh x and cosh x that

tanh^2 x = 1 - sech^2 x and

d/dx (tanh x) = sech^2 x

The question is to integrate (tanh x - tanh^3 (x)) which I could use a little help with.

I have got this into the form
sinh x / cosh^3 x or tanhx sech^2 x and tried integrating by parts but I got minus the exact same integral. If someone could just point out where I'm going wrong.

Thanks
tanhx differentiates to sech2x
tanh2x differentiates to 2 sech2xtanhx

does this help you?
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Hasufel
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follow TeeEm`s hint - what you are tring to integrate is:

(by using the trig identity: 1-tanh^2(x) = sech^2(x))

tanh(x)(1-tan^{2}h(x))= tanh(x) \times sech^{2}(x)

the hyperbolic integral result is the same as the non-hyperbolic (i.e. same functions but hyperbolic)
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Help_Me_Alex
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(Original post by TeeEm)
tanhx differentiates to sech2x
tanh2x differentiates to 2 sech2xtanhx

does this help you?
Yeah thanks, just as you posted that I did product rule on tanh^2 x and got it.
Thanks!
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TeeEm
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(Original post by Help_Me_Alex)
Yeah thanks, just as you posted that I did product rule on tanh^2 x and got it.
Thanks!
no worries
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