Question

Solve: $\int \frac{{\log }^x}{x^3}dx$

Answer 1

$\int \frac{\log x}{x^3}dx$

$=\int e^{-2t}tdt$ Let $\log x=t$ $\Rightarrow x=e^t\frac{dx}{x}=dt$

$=t\cdot \int e^{-2t}dt-\int \left(\frac{d\left(t\right)}{dt}\int e^{-2tdt}\right)dt$

$=\frac{t{e}^{-2t}}{(-2)}-\int \frac{1e^{-2t}}{(-2)}dt$

$=\frac{-t{e}^{-2t}}{2}-\frac{e^{-2t}}{4}+c$

$=\frac{e^{-2t}}{2}(-t-\frac{1}{2})+c$

$=\frac{e^{-2\log x}}{2}(-\log x-\frac{1}{2})+c$

$=\frac{1}{2x^2}(-\log x-\frac{1}{2})+c$ $(e^{\log x-2}=x^{-2})$

$\therefore \int \frac{\log x}{x^3}dx=\frac{1}{2x^2}(-\log x-\frac{1}{2})+c$.