# How do you calculate the antiderivative of #(-2x^3+14x^9)/(x^-2)#?

So, we want

Split this up, as we can split up sums or differences when integrating:

Factor out the constants:

So, integrating, we get

Yes, we would technically have two constants of integration as we had two integrals, but we absorb them all into one constant.

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To calculate the antiderivative of (-2x^3 + 14x^9)/(x^-2), you can rewrite the expression as -2x^5 + 14x^11, then integrate each term separately. The antiderivative of -2x^5 is -(2/6)x^6 = -(1/3)x^6, and the antiderivative of 14x^11 is (14/12)x^12 = (7/6)x^12. So, the antiderivative of (-2x^3 + 14x^9)/(x^-2) is -(1/3)x^6 + (7/6)x^12 + 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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