# How do you integrate #int x^5e^(-x^3)#?

We then have:

and undoing the substitution:

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To integrate ( \int x^5e^{-x^3} ), you can use substitution method. Let ( u = -x^3 ), then ( du = -3x^2dx ). Rearranging, ( dx = \frac{-1}{3x^2}du ). Substituting these into the integral gives ( \int x^5e^{-x^3}dx = \int x^2e^u \frac{-1}{3x^2}du ). Simplifying, ( \int x^5e^{-x^3}dx = \int \frac{-1}{3} e^u du ). Now, integrate ( \int \frac{-1}{3} e^u du ) with respect to ( u ) to get ( \frac{-1}{3}e^u + C ). Substitute back ( u = -x^3 ) to get the final result: ( \frac{-1}{3}e^{-x^3} + 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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