Tính \(\int\limits_0^a {x{{\left( {3 - x} \right)}^3}dx} \)

* Hướng dẫn giải

Đặt \(t = 3 - x \Rightarrow dt =  - dx\)

Khi \(x = 0\) thì \(t = 3\), khi \(x = a\) thì \(t = 3 - x\)

\(\begin{array}{l}
\int\limits_0^a {x{{\left( {3 - x} \right)}^3}dx}  =  - \int\limits_{3 - a}^3 {\left( {3{t^3} - {t^4}} \right)dt} \\
 = \left. {\left( {\frac{3}{4}{t^4} - \frac{1}{5}{t^5}} \right)} \right|_{3 - a}^3\\
 = \frac{{243}}{{20}} - \frac{3}{4}{\left( {a - 3} \right)^4} - \frac{1}{5}{\left( {a - 3} \right)^5}
\end{array}\)