How do you integrate #(x^2)(e^(4x))dx#?

To integrate ( \int x^2 e^{4x} , dx ), you can use integration by parts. Let ( u = x^2 ) and ( dv = e^{4x} , dx ). Then, ( du = 2x , dx ) and ( v = \frac{1}{4} e^{4x} ).

Using the integration by parts formula ( \int u , dv = uv - \int v , du ), we have: [ \int x^2 e^{4x} , dx = \frac{x^2}{4} e^{4x} - \frac{1}{4} \int 2x e^{4x} , dx ]

Now, we need to integrate ( \int 2x e^{4x} , dx ). This is again a product of functions, so we can apply integration by parts once more.

Let ( u = 2x ) and ( dv = e^{4x} , dx ). Then, ( du = 2 , dx ) and ( v = \frac{1}{4} e^{4x} ).

Using integration by parts again, we get: [ \int 2x e^{4x} , dx = \frac{2x}{4} e^{4x} - \frac{1}{4} \int 2 e^{4x} , dx ]

The integral ( \int 2 e^{4x} , dx ) is straightforward to evaluate.

Putting it all together, we have: [ \int x^2 e^{4x} , dx = \frac{x^2}{4} e^{4x} - \frac{1}{4} \left( \frac{2x}{4} e^{4x} - \frac{1}{4} e^{4x} \right) + C ]

Where ( C ) is the constant of integration.