missing $j*\omega$ in integral

Question

let us consider following integral

according to property of delta function,we can write this intgeral as

$\int^{t=\infty}_{t=t_0} e^{-j*\omega*t}$

or we can write as

$e^{-j*\omega*t}/(-\omega*t)$ from $t=t_0$ to $t=\infty$,if we calculate it we get

$\frac{e^{-j*w*t_0}} {w*j}$

but i did not understand why is not given in formula denominator part?thanks in advance

Answer 1

You might want to take a look at the answers to this question.

In less technical language: The delta “function” has the defining property that $$\int_{-\infty}^\infty \delta(x)f(x)\,dt=f(0)$$ for any continuous function $f$. Substituting in $x=t-t_0$ with $f(x)=e^{-j\omega x}$ immediately yields the desired result.

Your rewrite of the integral “according to property of delta function” is not according to any property of the delta known to me.