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Define J = integral of (sin²(x) + sin(x)) / (1 + sin(x) + cos(x)) dx and K = integral of (cos²(x) + cos(x)) / (1 + sin(x) + cos(x)) dx, where C is an arbitrary constant of integration. Which of the following relations is/are correct?

1. J = (1/2)*(x - sin x + cos x) + C
2. J = K - (sin x + cos x) + C
3. J = x - K + C
4. K = (1/2)*(x - sin x + cos x) + C

Correct answer: J = x - K + C

Solution

Adding J and K: numerator becomes sin² x + cos² x + sin x + cos x = 1 + sin x + cos x, which cancels the denominator, giving J + K = x + C. Therefore J = x - K + C, confirming option C.

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