I know that the CDF of a continuous uniform (over $[a,b]$) random variable is:
\begin{align} CDF_X(t) = \begin{cases} 0 &\text{if } t < a \\ \frac{t - a}{b - a} &\text{if } a \leq t < b \\ 1 &\text{if } t \geq b \end{cases} \end{align}
To get this result, is it ok to evaluate the 3 integrals below?
\begin{align} CDF_X(t) = \begin{cases} \displaystyle{\int \limits_{- \infty}^{t \leq a}} 0 \, dt = 0 \\ \displaystyle{\int \limits_{- \infty}^{t \leq b}} f(t) \, dt = \displaystyle{\int \limits_{- \infty}^{t \leq a}} 0 \, dt + \displaystyle{\int \limits_{a}^{t \leq b}} \frac{1}{b - a} \, dt = \frac{t - a}{b - a} \\ \displaystyle{\int \limits_{- \infty}^{t \leq + \infty}} f(t) \, dt = \displaystyle{\int \limits_{- \infty}^{t \leq a}} 0 \, dt + \displaystyle{\int \limits_{a}^{b}} \frac{1}{b - a} \, dt + \displaystyle{\int \limits_{b}^{t \leq + \infty}} 0 \, dt = 1 \end{cases} \end{align}