Use integration by parts to find ∫x e^(x)

So to integrate this function we would need to use a method called Integration by parts that you may have come across in your studies.
This is where we separate the function into two parts to make it easier to integrate.
We assign one part of the function the letter u, and the other part of the function the derivation of v , represented as dv.
We then differentiate the function assigned as u to find du and integrate the function assigned as dv to find v.
We then plug these values into the formula for integration by parts which is:
∫ = uv - ∫vdu
So in the case of this question we would assign the values:
u = x dv = e^(x)
We then differentiate x to get 1, and integrate e^x, which of course remains as e^(x).
du = 1 v = e^(x)
Plugging these into the equation we find:
∫ = x e^(x) - ∫e^(x)
The integral of e^x being e^x as we established giving us the formula
∫ = x e^(x) - e^(x)
Which can be simplified to:
∫ = e^(x) ( x - 1)
And as we integrated it is possible that there was a certain constant that disappeared when differentiated so we add + C to the end of the formula.
Giving us the end result of:
∫ x e^(x) = e^(x) ( x - 1 ) + C