7.1 Integration By Parts/51b

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

${\text{Exercise 47: }}x(\ln {x})^{n}-n\int (\ln {x})^{n-1}dx$

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{aligned}\int \ln(x)^{3}dx&=x\ln(x)^{3}-3\underbrace {\int \ln(x)^{2}dx} _{\begin{aligned}u&=\ln ^{2}{(x)}&dv&=dx\\[0.6ex]du&={\tfrac {2\ln {(x)}}{x}}dx&v&=x\end{aligned}}\\&=\int \ln(x)^{3}dx=x\ln(x)^{3}-3\end{aligned}}}