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Let y = f(x) solve the linear differential equation dy/dx + x*y/(x² - 1) = (x⁴ + 2x)/sqrt(1 - x²) on the interval (-1, 1), with f(0) = 0. Evaluate the integral of f(x) from x = -sqrt(3)/2 to x = sqrt(3)/2.
- pi/3 - sqrt(3)/2
- pi/3 - sqrt(3)/4
- pi/6 - sqrt(3)/4
- pi/6 - sqrt(3)/2
Correct answer: pi/3 - sqrt(3)/2
Solution
With IF sqrt(1-x²), the equation integrates to give f, an odd-plus-even combination; integrating over the symmetric interval yields pi/3 - sqrt(3)/2.
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