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Evaluate the integral ∫_(√(ln 3))^(√(ln 2)) (xsin(x²))/(sin(x²)+sin(ln 6-x²)) dx.

1. (1)/(4)ln ((3)/(2))
2. (1)/(2)ln ((3)/(2))
3. ln ((3)/(2))
4. (1)/(6)ln ((3)/(2))

Correct answer: (1)/(4)ln ((3)/(2))

Solution

Let u=x^2 (du=2x dx). The integral becomes (1/2)*Integral sin(u)/(sin(u)+sin(ln6-u)) du over [ln2,ln3]. Since ln2+ln3=ln6, adding the King-reflected integral gives (1/2)*(ln3-ln2)=(1/2)ln(3/2), so each half is (1/4)ln(3/2).

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