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Is the integral of delta function a scalar or a function u(t)?

Imagine integral of Dirac delta between minus infinity to plus infinity; or from minus infinity to a particular time x.

In some texts the integral of the delta is given as simply one and some it as u(t) unit step function.

Which one is correct? Im confused with the meaning of the derivation.

Bernard
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pnatk
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  • Do you know the difference between definite and indefinite integral? – Yuriy S Oct 28 '18 at 23:20
  • When does it become 1 and when does it become u(t)? If we integrate from minus infnity to x the result becomes u(x) which is 1 but they say it is u(t). Im very confused. – pnatk Oct 28 '18 at 23:23

1 Answers1

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$$\int_{-\infty}^t \delta(x) \, dx = \begin{cases} 1 & t \ge 0 \\ 0 & t < 0\end{cases} = u(t)$$ This is a function of $t$ because there is a variable $t$ in the upper limit of integration; for each value of $t$, there an integral to consider.

Taking $t \to \infty$ yields $\int_{-\infty}^\infty \delta(x) \, dx = 1$. This is a number because it is a single definite integral.

angryavian
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