- #1

evinda

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I want to calculate the integral $\int_{-1}^2\sin \left (\pi (t-1)\right )\delta (-t+1)\, dt$. I have done the following so far:

$$\int_{-\infty}^{+\infty}\sin \left (\pi (t-1)\right )\delta (-t+1)\, dt=\int_{-\infty}^1\sin \left (\pi (t-1)\right )\delta (-t+1)\, dt+\int_{-1}^2\sin \left (\pi (t-1)\right )\delta (-t+1)\, dt+\int_2^{+\infty}\sin \left (\pi (t-1)\right )\delta (-t+1)\, dt$$

We have that $\int_{-\infty}^{+\infty}\sin \left (\pi (t-1)\right )\delta (-t+1)\, dt=\int_{-\infty}^{+\infty}\sin \left (\pi (t-1)\right )\delta (t-1)\, dt=\sin (-\pi)=0$.

How can we continue? (Thinking) Do we use somehow the Heaviside step function? (Thinking)