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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