# What is the antiderivative of #x^3/(1+x^2)#?

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The antiderivative of (\frac{x^3}{1+x^2}) can be found using the substitution method. Let (u = 1 + x^2), then (du = 2x , dx). Rewriting the integral in terms of (u) gives (\int \frac{x^3}{1+x^2} , dx = \frac{1}{2} \int \frac{u-1}{u} , du). Simplifying and integrating gives (\frac{1}{2} \int \left(1 - \frac{1}{u}\right) , du = \frac{1}{2} (u - \ln|u|) + C = \frac{1}{2}(1 + x^2 - \ln(1 + x^2)) + C), where (C) is the constant of integration.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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