If f(x) is a continuous periodic function wit | vedclass

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(C) Let $g(a) = \int_a^{a + T} {f(x)dx}$.Differentiating with respect to $a$ using the Leibniz rule:$g'(a) = \frac{d}{da} \int_a^{a + T} {f(x)dx} = f(

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Independent of $a$

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(C) Let $g(a) = \int_a^{a + T} {f(x)dx}$.Differentiating with respect to $a$ using the Leibniz rule:$g'(a) = \frac{d}{da} \int_a^{a + T} {f(x)dx} = f(a + T) \cdot \frac{d}{da}(a + T) - f(a) \cdot \frac{d}{da}(a)$.Since $f(x)$ is a periodic function with period $T$,we have $f(a + T) = f(a)$.Thus,$g'(a) = f(a + T) - f(a) = f(a) - f(a) = 0$.Since the derivative $g'(a) = 0$,the function $g(a)$ is a constant value.Therefore,the integral $I = \int_a^{a + T} {f(x)dx}$ is independent of $a$.

If $f(x)$ is a continuous periodic function with period $T,$ then the integral $I = \int_a^{a + T} {f(x)\,dx} $ is

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