Find the integral of x^n e^x dx using integration by parts. What is the general form?

To find the integral of ( x^n e^x , dx ) using integration by parts, we first choose our functions to differentiate and integrate. We typically set ( u = x^n ) (which we will differentiate) and ( dv = e^x , dx ) (which we will integrate).

From this choice, we have:

• ( du = n x^{n-1} , dx )

• ( v = e^x )

Now applying the integration by parts formula, which states ( \int u , dv = u v - \int v , du ), we can substitute our values:

[

\int x^n e^x , dx = x^n e^x - \int e^x (n x^{n-1}) , dx

]

This simplifies to:

[

\int x^n e^x , dx = x^n e^x - n \int x^{n-1} e^x , dx

]

Rearranging gives us the expression for the integral:

[

\int x^n e^x , dx = x^n e^x -