7.1 Integration By Parts/51b: Difference between revisions
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\begin{aligned} | \begin{aligned} | ||
u & = \ln^{2}{(x)} & x &= \tfrac{1}{\sqrt{2}}(u+v) \\[0.6ex] | u & = \ln^{2}{(x)} & x &= \tfrac{1}{\sqrt{2}}(u+v) \\[0.6ex] | ||
du & = \tfrac{2}{\ln{(x)}}dx & y &= \tfrac{1}{\sqrt{2}}(u-v) | |||
\end{aligned} | \end{aligned} | ||
} | } | ||
Revision as of 17:49, 29 November 2022
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 \text{Use exercise 47 to evaluate} \int(\ln{x})^3dx }
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 \text{Exercise 47: } x(\ln{x})^n-n\int(\ln{x})^{n-1}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} \int\ln(x)^3dx &= x\ln(x)^3 -3\underbrace{\int\ln(x)^2dx}_{ \begin{aligned} u & = \ln^{2}{(x)} & x &= \tfrac{1}{\sqrt{2}}(u+v) \\[0.6ex] du & = \tfrac{2}{\ln{(x)}}dx & y &= \tfrac{1}{\sqrt{2}}(u-v) \end{aligned} } \end{align} }