The integral of \(\int \frac{dx}{x(x^4+1)}\) can be solved using the substitution \(u=x^2\), leading to the transformation of the integral into \(\frac{1}{2}\int \frac{du}{u^2+1}\). The correct evaluation results in \(\frac{1}{2}\arctan(x^2) + C\). However, a critical error was identified in the original approach, where the \(x\) factor in the denominator was not properly replaced, necessitating the use of partial fraction decomposition to correctly express the integral as \(\int \frac{A}{u} + \frac{Bu+C}{u^2+1} du\).
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The integral above isn't right. You forgot to replace the x factor in the denominator.nameVoid said:<br /> \int \frac{dx}{x(x^4+1)}<br />
<br /> u=x^2<br />
<br /> \sqrt{u}=x<br />
<br /> dx=\frac{1}{2\sqrt{u}}<br />
<br /> \frac{1}{2}\int \frac{du}{u^2+1}<br />
nameVoid said:<br /> \frac{1}{2}arctanx^2+C<br />