Integral producing indefinite answer

Question

I'm trying to solve an integral $$\int_0^tdt_2 e^{-\nu |t_1-t_2|}$$ and getting a result with a $\pm$, but this doesn't make sense physically. Where am I going wrong? First I use $u = t_2-t_1,du = dt_2$: \begin{align} I&=\int_{-t_1}^{t-t_1}dt_2\,e^{-\nu |u|} \end{align} Then use $v = |u|, \,dv = \frac{u}{|u|}du$; \begin{align} I = \int_{t_1}^{|t-t_1|}dv\frac{|u|}{u}e^{-\nu\,v} \end{align} which is where the problem comes in - to get rid of the $u$ term I'm forced to take abs$^{-1}[u] = \pm u$, giving $$I =\pm \int_{t_1}^{|t-t_1|}dv\,e^{-\nu\,v}$$ which is then easily solved, but only by having included this $\pm$. What am I missing?

Answer 1

Well I won't give you the full solution for the purposes of your own education, but consider the absolute value as a piecewise function

$$ |x| = \begin{cases} x , & x\geq0 \\ -x, & x<0\end{cases}$$

Then you can split the integral at the point where $t_2=t_1$ and solve them separately without dealing with the absolute value.