What is ∫ (x^3 e^x) dx using integration by parts?

To find the integral of ( \int x^3 e^x , dx ) using the method of integration by parts, we start with the formula:

[

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

]

For our integral, we can choose ( u = x^3 ) and ( dv = e^x , dx ). This means we need to differentiate ( u ) and integrate ( dv ):

• Differentiating ( u ):

[

du = 3x^2 , dx

]

• Integrating ( dv ):

[

v = e^x

]

Now we can apply the integration by parts formula:

[

\int x^3 e^x , dx = x^3 e^x - \int e^x \cdot 3x^2 , dx

]

Next, we need to compute the integral ( \int 3x^2 e^x , dx ). We will again use integration by parts on this integral. Set ( u = x^2 ) and ( dv = 3e^x , dx ):

• Differentiating ( u