Match derivatives in trigonometric composites

The Teacher scaffold is $f=2-3x$, $f^{\prime}=-3$. Thus $\int\sin(2-3x)\,dx=-\frac13\int(-3)\sin(2-3x)\,dx=\frac13\cos(2-3x)+C$.

With $f=3x$, $f^{\prime}=3$, $\int\sec(3x)\tan(3x)\,dx=\frac13\int3\sec(3x)\tan(3x)\,dx=\frac13\sec(3x)+C$, on intervals where $\cos(3x)\ne0$.

With $f=3x^2$, $f^{\prime}=6x$, $\int x\cos(3x^2)\,dx=\frac16\int6x\cos(3x^2)\,dx=\frac16\sin(3x^2)+C$.