Cho tích phân \(\int\limits_1^2 {\frac{{\ln x}}{{{...

D

- Hướng dẫn giải

\(I = \int\limits_1^2 {\frac{{\ln xdx}}{{{x^2}}}dx} \) 

Đặt \(\left\{ \begin{array}{l}
u = \ln x\\
dv = \frac{1}{{{x^2}}}dx
\end{array} \right. \Rightarrow \left\{ \begin{array}{l}
du = \frac{{dx}}{x}\\
v =  - \frac{1}{x}
\end{array} \right.\) ta có:

\(I = \left( {\ln x.\frac{{ - 1}}{x}} \right)\left| \begin{array}{l}
2\\
1
\end{array} \right. + \int\limits_1^2 {\frac{{dx}}{{{x^2}}} =  - \frac{1}{2}\ln 2 - \frac{1}{x}\left| \begin{array}{l}
2\\
1
\end{array} \right. =  - \frac{1}{2}\ln 2 - \frac{1}{2} + 1 = \frac{1}{2} - \frac{1}{2}\ln 2} \) 

\( \Rightarrow \left\{ \begin{array}{l}
b = 1\\
c = 2\\
a = \frac{{ - 1}}{2}
\end{array} \right. \Rightarrow P = 2a + 3b + c =  - 1 + 3 + 2 = 4.\)