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What's the integral of x^5/(x^2+1)?

The integral of x^5/(x^2+1) is (1/2)x^4 - (1/2)x^2 + (5/2)ln|x^2+1| + C.

To solve this integral, we can use polynomial division to rewrite the integrand as x^3 - x + (x^2 - 1)/(x^2 + 1). Then we can split the integral into two parts: the integral of x^3 - x, which can be easily evaluated using the power rule, and the integral of (x^2 - 1)/(x^2 + 1).

To evaluate the second integral, we can use the substitution u = x^2 + 1, du/dx = 2x, and rewrite the integral as (1/2)∫(u/x^2 - 1/x^2)du. The first term can be evaluated using another substitution v = u/x^2, dv/dx = -2u/x^3, and rewriting the integral as (-1/2)∫v dv. The second term can be evaluated using the natural logarithm rule.

Putting everything together, we get:

∫x^5/(x^2+1) dx = ∫(x^3 - x + (x^2 - 1)/(x^2 + 1)) dx
= (1/4)x^4 - (1/2)x^2 + (1/2)ln|x^2+1| - (1/2)∫(1 - 1/(x^2 + 1)) dx
= (1/4)x^4 - (1/2)x^2 + (5/2)ln|x^2+1| + C.

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