## Calculus 8th Edition

Published by Cengage

# Chapter 7 - Techniques of Integration - 7.1 Integration by Parts - 7.1 Exercises - Page 516: 19

#### Answer

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

#### Work Step by Step

$\int$udv = uv - $\int$vdu $\int$$z^{3}e^{z} u=z^{3} du=3z^{2}dz dv=e^{z}dz v=e^{z} \int$$z^{3}e^{z}$ = $z^{3}$$e^{z} - \int$$e^{z}$3$z^{2}$dz = $z^{3}$$e^{z} - 3\int$$e^{z}$$z^{2}dz Integration by parts again: u=z^{2} du=2zdz dv=e^{z}dz v=e^{z} z^{3}$$e^{z}$ - 3$\int$$e^{z}$$z^{2}$dz = $z^{3}$$e^{z} - 3(z^{2}$$e^{z}$ - $\int$$e^{z}$$2z$dz) = $z^{3}$$e^{z} - 3(z^{2}$$e^{z}$ - 2$\int$$e^{z}$$z$dz) Integration by parts one last time: u=$z$ du=dz dv=$e^{z}$dz v=$e^{z}$ $z^{3}$$e^{z} - 3(z^{2}$$e^{z}$ - 2$\int$$e^{z}$$z$dz) = $z^{3}$$e^{z} - 3(z^{2}$$e^{z}$ - 2($z$$e^{z}-\int$$e^{z}$dz)) = $z^{3}$$e^{z} - 3(z^{2}$$e^{z}$ - 2($z$$e^{z}-e^{z})) = z^{3}$$e^{z}$ - 3$z^{2}$$e^{z} - 6(z$$e^{z}$-$e^{z}$) = $z^{3}e^{z}-3z^{2}e^{z}+6ze^{z}-6e^{z}+C$

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