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Since Differentiation of a sinusoidal function of a certain angular frequency gives a sinusoidal function of the same frequency, does the statement "Integration of a sinusoidal function of certain frequency gives again a sinusoidal of same frequency" holds true or not?

I am asking this as I recall I have read that the second statement does not hold good but I cannot figure out now why.

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Yes and no.

If you recall from Calculus, if $g(t) = \frac{d}{dt} f(t)$, then $\int g(t)\ dt = f(t) + C$. So, since the derivative of a sinusoid is a sinusoid of the same frequency, if $g(t)$ sinusoidal, its integral must be a sinusoid of the same frequency, plus a constant.

Whether that "plus a constant" bit is going to make you consider the integral of a sinusoid to not be a sinusoid is up to you. Depending on the problem at hand, that constant term may or may not be a deal breaker for the "sinusoid-ness" of the result.

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    $\begingroup$ If we interpret "gives again a sinusoidal of same frequency" in a strict sense, yes. It indeed gives "a sinusoidal", plus something else $\endgroup$ Sep 18 at 10:13
  • $\begingroup$ Thanks, I was just wondering what might be the case that integrating a sinusoid giving a really different function. Yeah, I think so that context may not be treating DC component as a part of general sinusoidal signal. $\endgroup$ Sep 18 at 14:16

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