I have seen the Constant Multiple rule used in very confusing ways. The rule itself says that a constant in an integral can be moved out. $$\int kf(x) dx = k\int f(x) dx$$
However, this seems to lead to conflicting answers. I have seen people use it like so: $$\int kf(x) dx = \frac{1}{k}\int k\frac{1}{k}f(x) dx = \frac{1}{k}\int f(x) dx$$
Is this a leagal use of the rule? If so, when does this step apply?
I suspect the confusing use of the rule is actually \begin{align*} \int f(x) \,\mathrm{d}x &= \int 1 \cdot f(x) \,\mathrm{d}x && \text{always safe to multiply by $1$} \\ &= \int \left( \frac{1}{k} \cdot k \right) \cdot f(x) \,\mathrm{d}x && \text{valid for $k \neq 0$} \\ &= \frac{1}{k} \int k \cdot f(x) \,\mathrm{d}x && \text{constant multiple rule} \\ \end{align*}
This can be especially handy for integrands that you wish had a constant present. For instance, $f(x) = \mathrm{e}^{k x}$ would certainly be easier to antidifferentiate if that $k$ was there in the integrand. When we do this, it can seem that the $k$ disappears during the antidifferentiation -- it's better to think of the $k$ lubricating the application of the antidifferentation rule. Continuing the example, $$ \frac{1}{k}\int k \mathrm{e}^{kx} \,\mathrm{d}x = \frac{1}{k} \mathrm{e}^{kx} + C \text{.} $$