How to solve this integration $ \int_{0}^{\tau} e^{-|\Gamma-k/f|/\theta_1} d\Gamma $?

Question

I am trying to solve this expression.

$$ \int_{0}^{\tau} e^{-|\Gamma-k/f|/\theta_1} d\Gamma$$

$k,f,\theta_1$ are positive fixed variables.

Here, the is an absolute value of $\Gamma-k/f$. I tried to divide them in two parts and then I summed these two parts. But i failed. can any one help?

Answer 1

Let $a:= k/f$ then :

if $a \geq \tau$ then $\Gamma -a \leq 0$ hence : $$ \int_0^\tau e^{\frac{\Gamma-a}{\theta}} d\Gamma $$ if $a \leq 0$ then $\Gamma -a \geq 0$ hence : $$ \int_0^\tau e^{\frac{a-\Gamma}{\theta}} d\Gamma $$ else $$ \int_0^{a} e^{\frac{\Gamma-a}{\theta}} d\Gamma + \int_a^\tau e^{\frac{a-\Gamma}{\theta}} d\Gamma $$