Cho \(\int {2x{{\left( {3x - 2} \right)}^6}dx = A{{\left( {3x - 2} \right)}^8} + B{{\left( {3x - 2} \right)}^7} + C} \) với \(A,B,C \in R\).

* Hướng dẫn giải

\(I = \int {2x{{\left( {3x - 2} \right)}^6}dx} \) 

Đặt \(3x - 2 = t \Rightarrow x = \frac{{t + 2}}{3} \Rightarrow dx = \frac{1}{3}dt.\) 

\(\begin{array}{l}
 \Rightarrow I = \int {\frac{2}{9}\left( {t + 2} \right){t^6}dt = } \frac{2}{9}\int {\left( {{t^7} + 2{t^6}} \right)dt = \frac{2}{9}\left( {\frac{{{t^8}}}{8} + \frac{{2{t^7}}}{7}} \right) + C = \frac{1}{{36}}{t^8} + \frac{4}{{63}}{t^7} + C.} \\
 \Rightarrow I = \frac{1}{{36}}{\left( {3x - 2} \right)^8} + \frac{4}{{63}}{\left( {3x - 2} \right)^7} + C.\\
 \Rightarrow \left\{ \begin{array}{l}
A = \frac{1}{{36}}\\
B = \frac{4}{{63}}
\end{array} \right. \Rightarrow 12A + 7B = 12.\frac{1}{{36}} + 7.\frac{4}{{63}} = \frac{7}{9}.
\end{array}\)