# How do you find the antiderivative of #(4x^2 - 3x)e^x#?

The antiderivative of a function is its integral. Here, we need to solve:

Integrating by parts the integral, we get:

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To find the antiderivative of ( (4x^2 - 3x)e^x ), you can use integration by parts. Let ( u = 4x^2 - 3x ) and ( dv = e^x , dx ). Then, ( du = (8x - 3) , dx ) and ( v = e^x ).

Now, apply the integration by parts formula:

[ \int u , dv = uv - \int v , du ]

Substitute the values:

[ \int (4x^2 - 3x)e^x , dx = (4x^2 - 3x)e^x - \int e^x(8x - 3) , dx ]

Now, integrate ( e^x(8x - 3) ):

[ = (4x^2 - 3x)e^x - \int (8x - 3)e^x , dx ]

[ = (4x^2 - 3x)e^x - \left( 8 \int xe^x , dx - 3 \int e^x , dx \right) ]

[ = (4x^2 - 3x)e^x - \left( 8(xe^x - \int e^x , dx) - 3e^x \right) ]

[ = (4x^2 - 3x)e^x - \left( 8xe^x - 8e^x - 3e^x \right) ]

[ = (4x^2 - 3x)e^x - 8xe^x + 11e^x ]

[ = (4x^2 - 3x - 8x + 11)e^x ]

[ = (4x^2 - 11x + 11)e^x + C ]

So, the antiderivative of ( (4x^2 - 3x)e^x ) is ( (4x^2 - 11x + 11)e^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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