Let m, n, p, q be four positive integers. If | vedclass

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(D) Given $\int_0^{2 \pi} \sin^m x \cos^n x \, dx = 4 \int_0^{\pi/2} \sin^m x \cos^n x \, dx$. Since $\int_0^{2 \pi} f(x) \, dx = 4 \int_0^{\pi/2} f(x

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Both $a$ and $b$ are odd numbers

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(D) Given $\int_0^{2 \pi} \sin^m x \cos^n x \, dx = 4 \int_0^{\pi/2} \sin^m x \cos^n x \, dx$. Since $\int_0^{2 \pi} f(x) \, dx = 4 \int_0^{\pi/2} f(x) \, dx$ holds for even functions of $\sin$ and $\cos$ over quadrants,this implies $m$ and $n$ must be even.Given $\int_0^{2 \pi} \sin^p x \cos^n x \, dx = 0$. Since $n$ is even,$\cos^n x \ge 0$. For the integral to be $0$,$\sin^p x$ must be odd,so $p$ is odd.Given $\int_0^{\pi} \sin^p x \cos^q x \, dx = 0$. Since $p$ is odd,$\sin^p x$ is symmetric about $\pi/2$ but changes sign. For the integral to be $0$,$\cos^q x$ must be odd,so $q$ is odd.Now,$a = m + n + p = 	ext{even} + 	ext{even} + 	ext{odd} = 	ext{odd}$.And $b = m + n + q = 	ext{even} + 	ext{even} + 	ext{odd} = 	ext{odd}$.Thus,both $a$ and $b$ are odd numbers.

Let $m, n, p, q$ be four positive integers. If $\int_0^{2 \pi} \sin^m x \cos^n x \, dx = 4 \int_0^{\pi/2} \sin^m x \cos^n x \, dx$,$\int_0^{2 \pi} \sin^p x \cos^n x \, dx = 0$,$\int_0^{\pi} \sin^p x \cos^q x \, dx = 0$,$a = m + n + p$ and $b = m + n + q$,then:

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