How do you integrate $int x/(sqrt(1-x^4))$ using substitution?

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Answer 1 · 2016-09-12T19:04:38

$1/2arcsin(x^2)+C$

$intx/sqrt(1-x^4)dx$

Apply the substitution $x^2=sintheta$ . This implies that $2xdx=costhetad theta$ . Rearranging then substituting:

$=1/2int(2xdx)/sqrt(1-(x^2)^2)=1/2int(costhetad theta)/sqrt(1-sin^2theta)$

Note that $sqrt(1-sin^2theta)=costheta$ :

$=1/2intd theta$

$=1/2theta+C$

From $x^2=sintheta$ we see that $theta=arcsin(x^2)$ :

$=1/2arcsin(x^2)+C$

How do you integrate int x/(sqrt(1-x^4))int x/(sqrt(1-x^4)) using substitution?