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 (unknown function "\begin{aligned}"): {\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} }