Use exercise 47 to evaluate ∫ ( ln x ) 3 d x {\displaystyle {\text{Use exercise 47 to evaluate}}\int (\ln {x})^{3}dx} Exercise 47: x ( ln x ) n − n ∫ ( ln x ) n − 1 d x {\displaystyle {\text{Exercise 47: }}x(\ln {x})^{n}-n\int (\ln {x})^{n-1}dx}
∫ ln ( x ) 3 d x = x ln ( x ) 3 − 3 ∫ ln ( x ) 2 d x ⏟ u = ln 2 x d v = d x d u = 2 x ln ( x ) d x v = x {\displaystyle {\begin{aligned}\int \ln(x)^{3}dx&=x\ln(x)^{3}-3\underbrace {\int \ln(x)^{2}dx} _{\begin{aligned}u&=\ln ^{2}{x}\qquad dv=dx\\[2ex]du&={\frac {2}{x}}\ln {(x)}dx\quad v=x\\[2ex]\end{aligned}}\end{aligned}}}
![{\displaystyle {\begin{aligned}\int \ln(x)^{3}dx&=x\ln(x)^{3}-3\underbrace {\int \ln(x)^{2}dx} _{\begin{aligned}u&=\ln ^{2}{x}\qquad dv=dx\\[2ex]du&={\frac {2}{x}}\ln {(x)}dx\quad v=x\\[2ex]\end{aligned}}\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/05d310831dfbf1d5bb3c80bf082ba2a21713f8d5)