9. -/6.25 points details my notes evaluate the integral. (use c for the constant of integration.) ∫ 2z³e^z dz

Answer:

$2z^{3}e^{z}-6z^{2}e^{z}+12ze^{z}-12e^{z}+C$

Explanation:

Step1: Apply integration - by - parts formula

$\int u\mathrm{d}v=uv-\int v\mathrm{d}u$. Let $u = 2z^{3}$ and $\mathrm{d}v=e^{z}\mathrm{d}z$. Then $\mathrm{d}u = 6z^{2}\mathrm{d}z$ and $v = e^{z}$. So $\int 2z^{3}e^{z}\mathrm{d}z=2z^{3}e^{z}-\int 6z^{2}e^{z}\mathrm{d}z$.

Step2: Apply integration - by - parts again

For $\int 6z^{2}e^{z}\mathrm{d}z$, let $u = 6z^{2}$, $\mathrm{d}v=e^{z}\mathrm{d}z$. Then $\mathrm{d}u = 12z\mathrm{d}z$ and $v = e^{z}$. So $\int 6z^{2}e^{z}\mathrm{d}z=6z^{2}e^{z}-\int 12ze^{z}\mathrm{d}z$.

Step3: Apply integration - by - parts again

For $\int 12ze^{z}\mathrm{d}z$, let $u = 12z$, $\mathrm{d}v=e^{z}\mathrm{d}z$. Then $\mathrm{d}u = 12\mathrm{d}z$ and $v = e^{z}$. So $\int 12ze^{z}\mathrm{d}z=12ze^{z}-\int 12e^{z}\mathrm{d}z$.

Step4: Integrate the last integral

$\int 12e^{z}\mathrm{d}z = 12e^{z}+C$.

Step5: Substitute back

$\int 2z^{3}e^{z}\mathrm{d}z=2z^{3}e^{z}-(6z^{2}e^{z}-(12ze^{z}-12e^{z}))+C=2z^{3}e^{z}-6z^{2}e^{z}+12ze^{z}-12e^{z}+C$.