If $f \in \mathscr{R}(\alpha)$ on $[a, b]$ and if $c$ is a positive constant, then $f \in \mathscr{R}(c \alpha)$, and $$ \int_a^b f d (c \alpha) = c \int_a^b f d \alpha.$$
This is part of Theorem 6.12 (e) in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition:
My proof:
As $c > 0$ and as $\alpha$ is a monotonically increasing function on $[a, b]$, so $c \alpha$ is also monotonically increasing on $[a, b]$.
And, for any partition $P = \left\{ \ x_0, x_1, \ldots, x_n \ \right\}$ of $[a, b]$, we also have $$ L(P, f, c \alpha) = \sum_{i=1}^n \left( \inf_{x_{i-1} \leq x \leq x_i} f(x) \right) \left[ c \alpha\left( x_i \right) - c \alpha\left( x_{i-1} \right) \right] = c \sum_{i=1}^n \left( \inf_{x_{i-1} \leq x \leq x_i} f(x) \right) \left[ \alpha\left( x_i \right) - \alpha\left( x_{i-1} \right) \right] = c L(P, f, \alpha);$$ that is, $$ L(P, f, c \alpha) = c L(P, f, \alpha ) \ \mbox{ and similarly} \ U(P, f, c \alpha) = c U(P, f, \alpha). \tag{1} $$
Now as $f \in \mathscr{R}(\alpha)$ on $[a, b]$, so for every real number $\varepsilon > 0$ we can find a partition $P$ of $[a, b]$ such that $$ U(P, f, \alpha) - L(P, f, \alpha) < { \varepsilon \over c }, $$ and so from (1) we conclude that for this same partition $P$ of $[a, b]$, we have $$ U(P, f, c \alpha) - L(P, f, c \alpha) < \varepsilon, \tag{2} $$ from which it follows that $f \in \mathscr{R}( c \alpha)$.
Also from (1) and (2) we see that \begin{align} \int_a^b f d (c \alpha) &\leq U(P, f, c \alpha) \\ &< L(P, f, c \alpha ) + \varepsilon \\ &= c L(P, f, \alpha) + \varepsilon \\ &\leq c \int_a^b f d \alpha + \varepsilon \end{align} for every real number $\varepsilon > 0$, which implies that $$ \int_a^b f d (c \alpha) \leq c \int_a^b f d \alpha. \tag{A}$$
And, again from (1) and (2) we also have
\begin{align} c \int_a^b f d \alpha & \leq c U(P, f, \alpha) \\ &= U(P, f, c \alpha) \\ &< L(P, f, c \alpha) + \varepsilon \\ &\leq \int_a^b f d (c \alpha) + \varepsilon \end{align} for every real number $\varepsilon > 0$, which implies that $$ c \int_a^b f d \alpha \leq \int_a^b f d (c \alpha). \tag{B}$$From (A) and (B) the required result follows.
Is my proof satisfactory enough?