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 ⏟ c = 1 d d d = 12 ∫ ln ( x ) 3 d x = x ln ( x ) 3 − 3 ∫ ln ( x ) 2 d x {\displaystyle {\begin{aligned}\int \ln(x)^{3}dx&=x\ln(x)^{3}\underbrace {-3\int \ln(x)^{2}dx} \\{\begin{aligned}c&=1\\ddd&=12\\\end{aligned}}\int \ln(x)^{3}dx&=x\ln(x)^{3}-3\int \ln(x)^{2}dx\\\end{aligned}}}
