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Given that the integral from 0 to pi/2 of sin⁴(x) * cos²(x) dx equals pi/32, find the value of the integral from 0 to pi/4 of cos⁴(x) * sin²(x) dx.

1. 3*pi/32
2. 2*pi/32
3. pi/32
4. None of these

Correct answer: None of these

Solution

Let I = integral from 0 to pi/2 of sin⁴(x)*cos²(x) dx = pi/32. By substitution x -> pi/2 - x, integral from 0 to pi/2 of cos⁴(x)*sin²(x) dx = pi/32 as well. Now integral from 0 to pi/4 of cos⁴(x)*sin²(x) dx is only half the symmetric interval (0 to pi/2), but sin⁴*cos² and cos⁴*sin² are NOT symmetric about pi/4. The value works out to pi/64, which does not match any given option, so the answer is None of these.

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