# How do you evaluate the integral #int x^2e^x#?

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To evaluate the integral ∫x^2e^x, we use integration by parts. Let u = x^2 and dv = e^x dx. Then, du = 2x dx and v = e^x.

Using the integration by parts formula ∫u dv = uv - ∫v du, we have:

∫x^2e^x dx = x^2e^x - ∫2xe^x dx.

Now, we use integration by parts again for the second integral. Let u = 2x and dv = e^x dx. Then, du = 2 dx and v = e^x.

Applying the formula again, we get:

∫2xe^x dx = 2xe^x - ∫2e^x dx.

Now, we can evaluate the remaining integral:

∫2e^x dx = 2e^x.

Putting it all together, we have:

∫x^2e^x dx = x^2e^x - (2xe^x - 2e^x) = x^2e^x - 2xe^x + 2e^x + 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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