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Let ff and gg be continuous and differentiable, let gg have finite period pp, and let [a,b][a,b] be a finite closed interval. How many statements are always true? I.∫ab(f(x)−g(x)) dx=∫abf(x) dx−∫abg(x) dx\mathrm{I.}\quad\int_a^b(f(x)-g(x))\,dx=\int_a^bf(x)\,dx-\int_a^bg(x)\,dx II.∫0npg(x) dx=n∫0pg(x) dx\mathrm{II.}\quad\int_0^{np}g(x)\,dx=n\int_0^pg(x)\,dx III.lim⁡n→∞∫abf(x)g(nx) dx=1p(∫abf(x) dx)(∫0pg(x) dx)\mathrm{III.}\quad\lim_{n\to\infty}\int_a^bf(x)g(nx)\,dx=\frac1p\left(\int_a^bf(x)\,dx\right)\left(\int_0^pg(x)\,dx\right)

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