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 = x 2 + 1 d v = e − x d x d u = 2 x d x v = − e − x {\displaystyle {\begin{aligned}\int \ln(x)^{3}dx&=x\ln(x)^{3}-3\underbrace {\int \ln(x)^{2}dx} _{\begin{aligned}u&=x^{2}+1\quad dv=e^{-x}dx\\[2ex]du&=2xdx\qquad v=-e^{-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&=x^{2}+1\quad dv=e^{-x}dx\\[2ex]du&=2xdx\qquad v=-e^{-x}\\[2ex]\end{aligned}}\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/fd93c5bb2c81b43c06208af07cb1d0165284a20d)