What is the value of k ?
For the next two (02) items that follow : Let \( \int_0^{\frac{\pi}{2}} \frac{a \sin x + b \cos x}{(a+b)(\sin x + \cos x)} dx = k \)
- A. \frac{\pi}{4} ✓
- B. \frac{\pi}{2}
- C. \pi
- D. 2\pi
Correct Answer: A. \frac{\pi}{4}
Explanation
Using King's Rule (\( \int_0^a f(x) dx = \int_0^a f(a-x) dx \)), the integral transforms. Adding the original and transformed integrals yields \( 2k = \int_0^{\pi/2} \frac{(a+b)(\sin x + \cos x)}{(a+b)(\sin x + \cos x)} dx = \frac{\pi}{2} \). Thus, \( k = \frac{\pi}{4} \).
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