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∫_(-3π/2)^(-π/2) [(x + π)³ + cos²(x + 3π)] dx is equal to

1. π⁴/32
2. π⁴/32 + π/2
3. π/2
4. π/4 - 1

Correct answer: π/2

Solution

Let t=x+pi. Limits become -pi/2 to pi/2 and cos^2(x+3pi)=cos^2(t). Integral = int_{-pi/2}^{pi/2}(t^3+cos^2 t)dt. The t^3 term is odd -> 0, and int_{-pi/2}^{pi/2}cos^2 t dt = pi/2. Value = pi/2.

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