Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 7 - Techniques of Integration - 7.1 Integration by Parts - 7.1 Exercises - Page 517: 56


$$x^{4}e^x-4x^{3}e^x+12x^{2}e^x-24 \left[x e^x- e^x \right]+c$$

Work Step by Step

Given $$ \int x^{4}e^x d x$$ by using the form $$ \int x^{n} e^xd x = x^{n}e^x-n \int x^{n-1} e^xd x $$ \begin{align*} \int x^{4}e^x d x&=x^{4}e^x-4 \int x^{3} e^xd x\\ &=x^{4}e^x-4\left[x^{3}e^x-3 \int x^{2} e^xd x\right] \\ &=x^{4}e^x-4x^{3}e^x+12\left[x^{2}e^x-2 \int x e^xd x\right] \\ &=x^{4}e^x-4x^{3}e^x+12x^{2}e^x-24 \left[x e^x- e^x \right]+c \end{align*}
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