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## Homework Statement

Compute the integral

[tex]F(x) = \int^x_{-\infty} f(t) \;dt[/tex]

of the linear combination of Dirac delta-functions

[tex]f(t) = -2\delta (t) + \delta (t-1) + \delta (t-2)[/tex].

Express the result analytically (piecewise on a set of intervals) and draw a sketch of the function [tex]F(x)[/tex].

## The Attempt at a Solution

Does [tex]F(x) = -2H(x) + H(x-1) + H (x-2)[/tex] where H is the Heaviside function?

I know how to express the Heaviside/Delta functions in terms of 'jumps' in a graph but the actual values could be anything couldn't they? For instance:

[tex]\begin{displaymath} F(x) = \left\{ \begin{array}{lr}

0, & \;x \leq 0\\

-2, & \;0 < x \leq 1\\

-1, & \;1<x\leq 2\\

0, & \;x > 2

\end{array}

\right.[/tex]

and

[tex]\begin{displaymath} F(x) = \left\{ \begin{array}{lr}

1, & \;x \leq 0\\

-1, & \;0 < x \leq 1\\

0, & \;1<x\leq 2\\

1, & \;x > 2

\end{array}

\right.

\end{displaymath}[/tex]

both respresent that linear combination of Heaviside functions don't they?

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