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

1 Answer
Mind
Jan 28, 2015

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!

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