Biết \(\int\limits_2^4 {\dfrac{1}{{2x + 1}}\,dx = m\ln 5 + n\ln 3\,\left( {m,n \in R} \right)} \). Tính P = m – n .

* Hướng dẫn giải

Ta có:

\(\int\limits_2^4 \dfrac{1}{{2x + 1}}\,dx \)

\(= \dfrac{1}{2}\int\limits_2^4 \dfrac{1}{{2x + 1}}\,d\left( {2x + 1} \right) \)

\(= \dfrac{1}{2}\ln \left| {2x + 1} \right|\left| \begin{array}{l}{}^4\\_2\end{array} \right. \)

\(= \ln 3 - \dfrac{1}{2}\ln 5 = m\ln 5 + n\ln 3\, \)

Khi đó ta có: \(\left\{ \begin{array}{l}n = 1\\m =  - \dfrac{1}{2}\end{array} \right. \Rightarrow P = m - n =  - \dfrac{3}{2}.\)