What is the integral of 5z^3e^z ?

The integral of 5z^3e^z is 5(z^3e^z-3(z^2e^z-2(e^zz-e^z)))+C

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integral of 5z^3e^z

\int 5 z^{3} e^{z} d z

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

Solution steps

\int 5 z^{3} e^{z} d z

Take the constant out:

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

= 5 \cdot \int z^{3} e^{z} d z

Apply Integration By Parts

: z^{3} e^{z} - \int 3 z^{2} e^{z} d z

= 5 (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}))

= 5 (z^{3} e^{z} - 3 (z^{2} e^{z} - 2 (e^{z} z - e^{z})))

Add a constant to the solution

= 5 (z^{3} e^{z} - 3 (z^{2} e^{z} - 2 (e^{z} z - e^{z}))) + C

f (x) = 8 x^{3}, at x = \frac{1}{2}

arctan (cos (θ))

\frac{d y}{d x}, y = \frac{- 5 x}{5} + 3 x^{3} - \frac{4}{5 x}

y = 3 x^{2} - 5 x + 9

y^{^{''}} + 9 y = 3 sec (3 x)

What is the integral of 5z^3e^z ?
The integral of 5z^3e^z is 5(z^3e^z-3(z^2e^z-2(e^zz-e^z)))+C