What is \( \int_0^{\frac{\pi}{2}} \frac{a \cos x + b \sin x}{\sin x + \cos x} dx \) equal to ?
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. k
- B. 2k
- C. k(a + b) ✓
- D. \frac{k}{a+b}
Correct Answer: C. k(a + b)
Explanation
By applying King's Rule to the given expression, \( \int_0^{\pi/2} \frac{a \cos x + b \sin x}{\sin x + \cos x} dx \) is exactly the numerator portion. Since the given integral divides this by (a+b) to equal k, the integral without the denominator evaluates to k(a+b).
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