\begin{equation*}
15 \int_2^3 x \sqrt{x - 2} \: dx
\end{equation*}
\begin{equation*}
\begin{split}
&15 \int_2^3 x \sqrt{x - 2} \: dx \\\\
& 15 \int_2^3 (x - 2 + 2) \sqrt{x - 2} \: dx \\\\
& 15 \int_2^3 (x - 2) \sqrt{x - 2} + 2 \sqrt{x - 2} \: dx \\\\
& 15 \int_2^3 (x - 2) \sqrt{x - 2} \: \frac {d(x - 2)}{1} + 30 \int_2^3 \sqrt{x - 2} \: \frac {d(x - 2)}{1} \\\\
& 15 \int_2^3 (x - 2) \sqrt{x - 2} \: d(x - 2) + 30 \int_2^3 \sqrt{x - 2} \: d(x - 2) \\\\
& 15 \int_2^3 (x - 2)^{1 \frac 12} \: d(x - 2) + 30 \int_2^3 (x - 2)^{\frac 12} \: d(x - 2) \\\\
& 15 \:.\: \frac 25 \left [ (x - 2)^{2 \frac 12} \right]_2^3 + 30 \:.\: \frac 23 \left [(x - 2)^{1 \frac 12} \right]_2^3 \\\\
& 6 \left [ (x - 2)^2 \sqrt{x - 2} \right]_2^3 + 20 \left [(x - 2) \sqrt{x - 2} \right]_2^3 \\\\
& 6 \left [ (3 - 2)^2 \sqrt{3 - 2} - (2 - 2)^2 \sqrt{2 - 2} \right] + 20 \left [(3 - 2) \sqrt{3 - 2} - (2 - 2) \sqrt{2 - 2} \right] \\\\
& 6 \left [ 1 - 0 \right] + 20 \left [1 - 0 \right] \\\\
& 26
\end{split}
\end{equation*}