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 ) x = 1 2 ( u + v ) d u = 2 ln ( x ) d x y = 1 2 ( u − v ) {\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)}&x&={\tfrac {1}{\sqrt {2}}}(u+v)\\[0.6ex]du&={\tfrac {2}{\ln {(x)}}}dx&y&={\tfrac {1}{\sqrt {2}}}(u-v)\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)}&x&={\tfrac {1}{\sqrt {2}}}(u+v)\\[0.6ex]du&={\tfrac {2}{\ln {(x)}}}dx&y&={\tfrac {1}{\sqrt {2}}}(u-v)\end{aligned}}\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/718f729587f5d8b9543d2ba35748e61971bbeeaf)