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Which of the following integral identities are true? (P) Integral from a to (pi-a) of x*f(sin x) dx = (pi/2) * Integral from a to (pi-a) of f(sin x) dx (Q) Integral from 0 to n*pi of f(cos²(x)) dx = n * Integral from 0 to pi of f(cos²(x)) dx (R) Integral from -a to a of f(x²) dx = 2 * Integral from 0 to a of f(x²) dx (S) Integral from 0 to (b-c) of f(x+c) dx = Integral from c to b of f(x) dx

1. Integral from a to (pi-a) of x*f(sin x) dx = (pi/2) * Integral from a to (pi-a) of f(sin x) dx
2. Integral from 0 to n*pi of f(cos²(x)) dx = n * Integral from 0 to pi of f(cos²(x)) dx
3. Integral from -a to a of f(x²) dx = 2 * Integral from 0 to a of f(x²) dx
4. Integral from 0 to (b-c) of f(x+c) dx = Integral from c to b of f(x) dx

Correct answer: Integral from 0 to (b-c) of f(x+c) dx = Integral from c to b of f(x) dx

Solution

All four statements are actually true. (P) is the generalized king's property. (Q) follows from periodicity of cos²(x) with period pi. (R) uses the even symmetry of f(x²). (S) is a simple substitution u=x+c. However, (P) requires careful check of the substitution on the non-symmetric interval [a, pi-a] — it does hold by the property integral of x*g(x) over [a,b] where g is symmetric about (a+b)/2. All four are correct.

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