7.1 Integration By Parts/48: Difference between revisions
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\int_{}^{} \left(x^{n} e^{x} \right)dx | Prove | ||
<math> | |||
\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> | |||
\begin{align} | |||
u &= x^{n} \quad & dv= e^{x} dx \\[2ex] | |||
du &=n x^{n-1} dx \quad & v=e^x \\[2ex] | |||
\end{align} | |||
</math> | |||
<math> | |||
\begin{align} | |||
\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> | |||
Latest revision as of 16:48, 29 November 2022
Prove Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \int_{}^{} \left(x^{n} e^{x} \right)dx = x^{n} e^{x} - n\int_{}^{} \left(x^{n-1} e^{x}\right)dx }
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \int_{}^{} \left(x^{n} e^{x} \right)dx }
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} u &= x^{n} \quad & dv= e^{x} dx \\[2ex] du &=n x^{n-1} dx \quad & v=e^x \\[2ex] \end{align} }
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \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} }