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How to integrate x/(1+x^4)?

To integrate x/(1+x^4), use the substitution u = x^2.

To integrate x/(1+x^4), use the substitution u = x^2. Then, du/dx = 2x and dx = du/(2x). Substituting these into the integral gives:

∫x/(1+x^4) dx = 1/2 ∫u/(1+u^2) du

To evaluate this integral, use another substitution v = 1+u^2, then dv/du = 2u and du = dv/(2u). Substituting these into the integral gives:

1/2 ∫u/(1+u^2) du = 1/4 ∫1/(1+u^2) dv

This integral can be evaluated using the inverse tangent function:

1/4 ∫1/(1+u^2) dv = 1/4 tan^-1(u) + C

Substituting back in for u gives:

∫x/(1+x^4) dx = 1/4 tan^-1(x^2) + C

Therefore, the integral of x/(1+x^4) is 1/4 tan^-1(x^2) + C.

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