I'm trying to find out which integration trick is going to help me with this integral: $$ \int_{t}^{\infty} \exp(- |x - a|) \mathrm{\ d}x $$ where $t, a \in \mathbb{R}$.
Solving an indefinite integral with the absolute value is easy, just splitting into two parts and solve the sum $\int_{-\infty}^{a}\ldots + \int_{a}^{\infty}\ldots$
But I'm afraid that the parameter $t$ makes things more complicated (such as having two solutions, depending on whether $t < a$ or not). Or is there any substitution or another trick?
First make a substitution of $x-a=u$ to get that the integral is: $$\int_{t-a}^\infty e^{-|u|}du$$If $t-a$ is greater than $0$ then integrate $e^{-u}$ like a regular integral. But if $t-a<0$ then calculate the following: $$\int^{0}_{t-a}e^{-u}du+\int_{t-a}^\infty e^{-u}du$$The logic behind these conditions is the fact that there is a line of symmetry at $u=a$.