7.1 Integration By Parts/51b

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

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

${\begin{aligned}\int \ln(x)^{3}dx&=x\ln(x)^{3}-3\underbrace {\int \ln(x)^{2}dx} _{\begin{aligned}u&={\tfrac {1}{\sqrt {2}}}(x+y)\qquad &x&={\tfrac {1}{\sqrt {2}}}(u+v)\\[0.6ex]v&={\tfrac {1}{\sqrt {2}}}(x-y)\qquad &y&={\tfrac {1}{\sqrt {2}}}(u-v)\end{aligned}}\end{aligned}}$