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 "\dx"): {\displaystyle \begin{align} \int\ln(x)^3dx &= x\ln(x)^3 -\underbrace{3\int\ln(x)^2dx}_{ \begin{aligned} u & = \ln^{2}{(x)} & dv &= dx \\[0.6ex] du & = \tfrac{2\ln{(x)}}{x}dx & v &= x \end{aligned} } \\ [1ex] &= x\ln(x)^3 -3\left[\ln^{2}{(x)}\cdot x - 2\int\ln{(x)}dx\right] \\ [1ex] &= x\ln(x)^3 -3x\ln^{2}{(x)} + \underbrace{6\int\ln{(x)}dx}_{ \begin{aligned} u & = \ln{(x)} & dv &= dx \\[0.6ex] du & = \tfrac{1}{x}dx & v &= x \end{aligned}} \\ [1ex] &= x\ln(x)^3 -3x\ln^{2}{(x)} + 6\left[\ln{(x)}\cdot x - \int\dx\right] \end{align} }