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Gate 2016 EE (Set 1) Q9

Gate 2016 EE (Set 1) Q9

9. The value of \int_{-\infty}^{+\infty} e^(-t) \delta (2t-2)dt  where  \delta (t) is the Dirac delta function, is:--------

Solution:

Using sifting property of Impulse function,

\int_{-\infty }^{+\infty }f(t)\delta (t-t_0 )dt=f(t)|t=t_0

\int_{-\infty }^{+\infty }e^(-1) \delta (2t-t)dt=> \int_{-\infty }^{+\infty }e^(-t) (1/2)\delta (t-1)dt att_0=1

e^(-t)/2 at t=1=>e^(-1)/2=>1/2e

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