^{2\pi} \sin^m x \cos^n x \, dx $.

Since the given condition is:

$ I = 4 \int_{0}^{\pi} \sin^m x \cos^n x \, dx $

We know that:

$ \int_{-a}^{a} f(x) \, dx = 2 \int_{0}^{a} f(x) \, dx $

if $ f(x) $ is an even function. This suggests that $ \sin^m x \cos^n x $ might be even in this case. To satisfy the condition where the integral over the larger symmetric interval is four times the integral over the smaller positive interval, $ m $ must be even.

Similarly, consider:

$ \int_{-\pi}^{\pi} \sin^r x \cos^s x \, dx = 4 \int_{0}^{\pi/2} \sin^r x \cos^s x \, dx $

Here, for a comparable reason, we must have $ r $ as even.

Finally, for:

$ \int_{-\pi/2}^{\pi/2} \sin^l x \cos^m x \, dx = 0 $

the function should be odd because the integral of an odd function over a symmetric interval around zero results in zero. Thus, $ l $ must be odd.

With the constraints $ 9 > m > l > s > n > r > 2 $, and that $ m $ and $ r $ are even while $ l $ is odd, a possible assignment of these integers could be:

$ m = 8 $

$ l = 7 $

$ s = 6 $

$ n = 5 $

$ r = 4 $

Next, let's test the option that matches this setup:

For option C, $(s-2)(s+2) = ln - 3$:

$ (s-2)(s+2) = (6-2)(6+2) = 4 \times 8 = 32 $

The expression $ ln - 3 $ was a typo originally intended to test our guess, but $ 32 = 32 $ checks out, confirming this is our desired setup.

Definite Integration

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