What is the integral of 3z^3e^z ?

The integral of 3z^3e^z is 3(z^3e^z-3(z^2e^z-2(e^zz-e^z)))+C

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integral of 3z^3e^z

\int 3 z^{3} e^{z} d z

3 (z^{3} e^{z} - 3 (z^{2} e^{z} - 2 (e^{z} z - e^{z}))) + C

Solution steps

\int 3 z^{3} e^{z} d z

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 3 \cdot \int z^{3} e^{z} d z

Apply Integration By Parts

= 3 (z^{3} e^{z} - \int 3 z^{2} e^{z} d z)

\int 3 z^{2} e^{z} d z = 3 (z^{2} e^{z} - 2 (e^{z} z - e^{z}))

= 3 (z^{3} e^{z} - 3 (z^{2} e^{z} - 2 (e^{z} z - e^{z})))

Add a constant to the solution

= 3 (z^{3} e^{z} - 3 (z^{2} e^{z} - 2 (e^{z} z - e^{z}))) + C

What is the integral of 3z^3e^z ?
The integral of 3z^3e^z is 3(z^3e^z-3(z^2e^z-2(e^zz-e^z)))+C