Answer
$$x{\ln ^4}x - 4x{\ln ^3}x + 12x{\ln ^2}x - 24x\ln x + 24x + C$$
Work Step by Step
$$\eqalign{
& \int {{{\ln }^4}x} dx \cr
& {\text{Using the reduction formula }} \cr
& \int {{{\ln }^n}xdx} = x{\ln ^n}x - n\int {{{\ln }^{n - 1}}xdx} \cr
& {\text{then,}} \cr
& \int {{{\ln }^4}x} dx = x{\ln ^4}x - 4\int {{{\ln }^{4 - 1}}xdx} \cr
& \int {{{\ln }^4}x} dx = x{\ln ^4}x - 4\int {{{\ln }^3}xdx} \cr
& {\text{Using the reduction formula }} \cr
& \int {{{\ln }^4}x} dx = x{\ln ^4}x - 4\left( {x{{\ln }^3}x - 3\int {{{\ln }^2}xdx} } \right) \cr
& \int {{{\ln }^4}x} dx = x{\ln ^4}x - 4x{\ln ^3}x + 12\int {{{\ln }^2}xdx} \cr
& {\text{Using the reduction formula }} \cr
& \int {{{\ln }^4}x} dx = x{\ln ^4}x - 4x{\ln ^3}x + 12\left( {x{{\ln }^2}x - 2\int {{{\ln }^{2 - 1}}xdx} } \right) \cr
& \int {{{\ln }^4}x} dx = x{\ln ^4}x - 4x{\ln ^3}x + 12x{\ln ^2}x - 24\int {\ln xdx} \cr
& \int {{{\ln }^4}x} dx = x{\ln ^4}x - 4x{\ln ^3}x + 12x{\ln ^2}x - 24\left( {x\ln x - \int {dx} } \right) \cr
& \int {{{\ln }^4}x} dx = x{\ln ^4}x - 4x{\ln ^3}x + 12x{\ln ^2}x - 24x\ln x + 24x + C \cr} $$
