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Let f(x) = integral from 0 to x of sin⁴(t) dt. Express f(x + pi) in terms of f(x) and f(pi).

1. f(pi)
2. f(x)
3. f(x) + f(pi)
4. f(x) * f(pi)

Correct answer: f(x) + f(pi)

Solution

f(x+pi) = integral₀^(x+pi) sin⁴(t) dt. Split: = integral₀^pi sin⁴(t) dt + integral_pi^(x+pi) sin⁴(t) dt. In the second integral, let u = t - pi: dt = du, sin(t) = sin(u+pi) = -sin(u), sin⁴(t) = sin⁴(u). Limits: u=0 to x. So second integral = integral₀^x sin⁴(u) du = f(x). Thus f(x+pi) = f(pi) + f(x).

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