7.1 Integration By Parts/51

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

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

$\int (\ln {x})^{3}dx=x(\ln {x})^{3}-3\int (\ln {x})^{2}dx$

$u=(\ln {x})^{2}\qquad dv=dx$

$du=2{\frac {1}{x}}\ln {x}dx\qquad v=x$

${\begin{aligned}\int (\ln {x})^{3}dx&=x(\ln {x})^{3}-3\int (\ln {x})^{2}dx=x(\ln {x})^{3}-3[x(\ln {x})^{2}-2\int (\ln {x})dx]\\[2ex]&=x(\ln {x})^{3}-3[x(\ln {x})^{2}-2(x\ln {x}-\int 1dx)=x(\ln {x})^{3}-3[x(\ln {x})^{2}-2x\ln {x}+2x]\\[2ex]&=x(\ln {x})^{3}-3x(\ln {x})^{2}+6x\ln {x}-6x+c\\[2ex]\end{aligned}}$