How do you find the integral of $tanh^3x dx$ ?

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Answer 1 · 2018-08-12T15:29:29

$inttanh(x)^3dx=ln(|cosh(x)|)+1/2sech(x)^2+C$ , $C in RR$

$I=inttanh(x)^3dx$

$=intsinh(x)^3/(cosh(x)^3)dx$

Because $sinh(x)^2=cosh(x)^2-1$ ,

$I=int(sinh(x)(cosh(x)^2-1))/(cosh(x)^3)dx$

Now let $u=cosh(x)$

$du=sinh(x)dx$

So:

$I=int(u^2-1)/u^3du$

$=int1/udu-int1/u^3du$

$=ln(|u|)+1/(2u^2)+C$ , $C in RR$

$=ln(|cosh(x)|)+1/2sech(x)^2+C$ , $C in RR$

\0/ Here's our answer !

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