Cho \(F\left( x \right) =  - \frac{1}{{3{x^3}}}\) ...

C

- Hướng dẫn giải

Ta có: \(F'(x) = \frac{1}{3}.\frac{{3{x^2}}}{{{x^6}}} = \frac{1}{{{x^4}}} = \frac{{f\left( x \right)}}{x} \Rightarrow f\left( x \right) = \frac{1}{{{x^3}}}\)

Xét \(I = \int {f'\left( x \right)\ln x} {\rm{d}}x\). Đặt \(\left\{ \begin{array}{l}
u = \ln x\\
{\rm{d}}v = f'\left( x \right){\rm{d}}x
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
{\rm{d}}u = \frac{1}{x}{\rm{d}}x\\
v = f\left( x \right)
\end{array} \right.\).

Ta có: \(I = \ln x.f\left( x \right) - \int {\frac{{f\left( x \right)}}{x}{\rm{d}}x + C = \frac{{\ln x}}{{{x^3}}} + \frac{1}{{3{x^3}}} + C} \).