University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson

Chapter 8 - Section 8.1 - Integration by Parts - Exercises - Page 427: 15

Answer

$$\int x^3e^xdx=x^3e^x-3x^2e^x+6xe^x-6e^x+C$$

Work Step by Step

$$A=\int x^3e^xdx$$ Set $u=x^3$ and $dv=e^xdx$ Then we have $du=3x^2dx$ and $v=e^x$ Using the formula $\int udv= uv-\int vdu$: $$A=x^3e^x-3\int x^2e^xdx$$ Set $u=x^2$ and $dv=e^xdx$ Then we have $du=2xdx$ and $v=e^x$ Using the formula $\int udv= uv-\int vdu$: $$A=x^3e^x-3\Big(x^2e^x-2\int xe^xdx\Big)$$ Set $u=x$ and $dv=e^xdx$ Then we have $du=dx$ and $v=e^x$ Using the formula $\int udv= uv-\int vdu$: $$A=x^3e^x-3\Bigg(x^2e^x-2\Big(xe^x-\int e^xdx\Big)\Bigg)$$ $$A=x^3e^x-3\Bigg(x^2e^x-2\Big(xe^x-e^x\Big)\Bigg)+C$$ $$A=x^3e^x-3\Bigg(x^2e^x-2xe^x+2e^x\Bigg)+C$$ $$A=x^3e^x-3x^2e^x+6xe^x-6e^x+C$$

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