Question

How does integration by parts work?

Answer 1

Integration by Parts is like the product rule for integration, in fact, it is derived from the product rule for differentiation. It states
`int u dv =uv-int v du`.

Let us look at the integral
`int xe^x dx`.

Let `u=x`.
By taking the derivative with respect to `x`
`Rightarrow {du}/{dx}=1`
by multiplying by `dx`,
`Rightarrow du=dx`

Let `dv=e^xdx`.
By dividing by `dx`
`Rightarrow {dv}/{dx}=e^x`
by integrating,
`Rightarrow v=e^x`

Now, by Integration by Parts,
#int xe^xdx =xe^x-inte^xdx=xe^x-e^x+C#