I've been trying to determine the integral $$\int_{0}^{\tau_2} \delta(\tau_1)e^{\tau_1a_2} \;\mathrm{d}\tau_1$$
By the sifting property, I would assume the answer to be $1$.
However, when using MATLAB's integral calculator this gives a value $\frac{sgn(\tau_1)}{2}$. To further confuse matters, WolframAlpha's online integral calculator gives $H(\tau_1)$, where $H$ is the Heaviside step function.
Which of these is correct? And why are the three methods giving different results? If anyone can shed light on this problem, that would be great. Thanks in advance!