Question

How do I evaluate `int(z +1) e^(3z) dz`?

Answer 1

`int (z+1)e^(3z) dz = int z*e^(3z)dz + int e^(3z)dz`

Second component is trivial
`int e^(3z) dz = int e^(3z)/3 d(3z) = e^(3z)/3`

For the first component, we use integration by part:
Recall: Integration by part: `int u(dv) = uv - int v(du)`
let `u = z` and `dv = e^(3z)`.
If `dv = e^(3z)`, then `v = e^(3z)/3`
(from second component)

Thus, `int z*e^(3z) dz = z*e^(3z)/3 - int e^(3z) dz`

Put the two component together:

`int (z+1)e^(3z) dz`
`= int z*e^(3z) dz + int e^(3z)dz`
`= [ z*e^(3z)/3 - int e^(3z) dz] + int e^(3z)dz`
`= z*e^(3z)/3 + C`

Hope this help!