How do you integrate #int x^3 e^(4x) dx # using integration by parts?
Use integration by parts three times.
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To integrate ( \int x^3 e^{4x} , dx ) using integration by parts:

Choose ( u ) and ( dv ): Let ( u = x^3 ) and ( dv = e^{4x} , dx ).

Compute the differentials: ( du = 3x^2 , dx ) and ( v = \frac{1}{4}e^{4x} ).

Apply the integration by parts formula: [ \int u , dv = uv  \int v , du ]
Substituting the values: [ \int x^3 e^{4x} , dx = x^3 \left( \frac{1}{4}e^{4x} \right)  \int \left( \frac{1}{4}e^{4x} \right) (3x^2 , dx) ]

Simplify and integrate the remaining integral: [ = \frac{1}{4}x^3 e^{4x}  \frac{3}{4} \int x^2 e^{4x} , dx ]
Now, you can use integration by parts again for the second integral.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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