Giải thích các bước giải:
Ta có:
\(\begin{array}{l}
x = a + b - t \Rightarrow \left\{ \begin{array}{l}
dx = - dt\\
x = a \Rightarrow t = b\\
x = b \Rightarrow t = a
\end{array} \right.\\
\Rightarrow \int\limits_a^b {x.f\left( x \right)dx} = \int\limits_b^a {\left( {a + b - t} \right).f\left( {a + b - t} \right).\left( { - dt} \right)} \\
= - \int\limits_a^b {\left( {a + b - t} \right)f\left( {a + b - t} \right)\left( { - dt} \right)} \\
= \int\limits_a^b {\left( {a + b - t} \right)f\left( {a + b - t} \right)dt} \\
= \int\limits_a^b {\left( {a + b - t} \right)f\left( t \right)dt\,\,\,\,\,\,\,\left[ {f\left( {a + b - t} \right) = f\left( t \right)} \right]} \\
= \int\limits_a^b {\left( {a + b} \right)f\left( t \right)dt} - \int\limits_a^b {tf\left( t \right)dt} \\
= \int\limits_a^b {\left( {a + b} \right)f\left( x \right)dx} - \int\limits_a^b {xf\left( x \right)dx} \\
\Rightarrow 2\int\limits_a^b {xf\left( x \right)dx} = \int\limits_a^b {\left( {a + b} \right)f\left( x \right)dx} = \left( {a + b} \right).\int\limits_a^b {f\left( x \right)dx} \\
\Rightarrow \int\limits_a^b {xf\left( x \right)dx} = \frac{{a + b}}{2}\int\limits_a^b {f\left( x \right)dx}
\end{array}\)
