7.1 Integration By Parts/48: Difference between revisions
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Prove | |||
<math> | <math> | ||
\int_{}^{} \left(x^{n} e^{x} \right)dx = x^{n} e^{x} - n\int_{}^{} \left(x^{n-1} e^{x}\right)dx | \int_{}^{} \left(x^{n} e^{x} \right)dx = x^{n} e^{x} - n\int_{}^{} \left(x^{n-1} e^{x}\right)dx | ||
</math> | |||
<math> | |||
\int_{}^{} \left(x^{n} e^{x} \right)dx | |||
</math> | </math> | ||
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\begin{align} | \begin{align} | ||
u &= x^{n} \quad dv= e^{x} dx \\[2ex] | u &= x^{n} \quad & dv= e^{x} dx \\[2ex] | ||
du &=n x^{n-1} dx \quad v=e^x \\[2ex] | du &=n x^{n-1} dx \quad & v=e^x \\[2ex] | ||
\end{align} | \end{align} | ||
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<math> | <math> | ||
\begin{align} | |||
\int_{}^{} \left(x^{n} e^{x} \right)dx = n^{n}e^{x} - \int_{}^{} \left(x^{n-1}e^{x}\right)dx | \int_{}^{} \left(x^{n} e^{x} \right)dx &= x^{n}e^{x} - \int_{}^{} \left(n x^{n-1}e^{x}\right)dx \\[2ex] | ||
&= x^{n}e^{x} - n \int_{}^{} \left(x^{n-1}e^{x}\right)dx \\[2ex] | |||
\end{align} | |||
</math> | </math> | ||
Latest revision as of 16:48, 29 November 2022
Prove
![{\displaystyle {\begin{aligned}u&=x^{n}\quad &dv=e^{x}dx\\[2ex]du&=nx^{n-1}dx\quad &v=e^{x}\\[2ex]\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/8cc421691d2243b55bbe0a0fd3e1babcb1049981)
![{\displaystyle {\begin{aligned}\int _{}^{}\left(x^{n}e^{x}\right)dx&=x^{n}e^{x}-\int _{}^{}\left(nx^{n-1}e^{x}\right)dx\\[2ex]&=x^{n}e^{x}-n\int _{}^{}\left(x^{n-1}e^{x}\right)dx\\[2ex]\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/b7548e0ad0dcf7796ae2f6e505141f27e2ea8399)