# What is the antiderivative of #x/(1+x^4) #?

Hence

now the integral becomes

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To find the antiderivative of ( \frac{x}{1+x^4} ), we can use a substitution. Let ( u = x^2 ). Then, ( du = 2x , dx ).

Rewriting the integral in terms of ( u ), we get:

[ \int \frac{x}{1+x^4} , dx = \frac{1}{2} \int \frac{2x}{1+x^4} , dx = \frac{1}{2} \int \frac{du}{1+u^2} ]

Now, we can use the formula ( \int \frac{du}{1+u^2} = \arctan(u) + C ), where ( C ) is the constant of integration. Substituting back ( u = x^2 ), we have:

[ \frac{1}{2} \int \frac{du}{1+u^2} = \frac{1}{2} \arctan(u) + C = \frac{1}{2} \arctan(x^2) + C ]

So, the antiderivative of ( \frac{x}{1+x^4} ) is ( \frac{1}{2} \arctan(x^2) + C ), where ( C ) is the constant of integration.

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