7.1 Integration By Parts/51b

${\text{Use exercise 47 to evaluate}}\int (\ln {x})^{3}dx$

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 \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] v & = \tfrac{2}{\ln{(x)}dx & y &= \tfrac{1}{\sqrt{2}}(u-v) \end{aligned} } \end{align} }