7.1 Integration By Parts/51b

Failed to parse (syntax error): {\displaystyle \text{Use } \int(\ln{(x)}^{n} = x(\ln{x})^n - n\int(\ln{x})^{n-1}dx \\ \text{ to evaluate } \int(\ln{x})^3dx}

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