paradigm 0
\[\begin{align}
\int x\cos xdx=?
\\ \\
\int x(\sin x)^{\prime}dx
\\ \\
設:u=x, \quad q=\sin x
\\ \\
\Rightarrow\int x\left(dx\cdot\sin x\right)=
\int x d\left(\sin x\right)
\\ \\
=x\sin x-\int\sin xdx
\\ \\
=x\sin x-(-\cos x)+C
\\ \\
=x\sin x+\cos x+C
\end{align}
\]
paradigm 1
\[\begin{align}
\int xe^xdx=?
\\ \\
\because (e^{x})^{\prime}=e^{x}\ln e=e^{x}
\\ \\
\Rightarrow\int xd(e^{x})=xe^{x}-\int e^{x}dx
\\ \\
=xe^{x}-(e^{x}+C)
\\ \\
=e^{x}\left(x-1\right)+C
\end{align}
\]
paradigm 2
\[\int x^{2} \ln x dx=?
\]