What is the value of \alpha?
Consider the following for the next two (02) items that follow : Given that \int\frac{3 \cos x+4 \sin x}{2 \cos x+5 \sin x}\,dx=\frac{\alpha x}{29}+\frac{\beta}{29}\ln|2 \cos x+5 \sin x|+c
- A. 7
- B. 13
- C. 17
- D. 26 ✓
Correct Answer: D. 26
Explanation
Express the numerator as A(\text{denominator}) + B(\text{derivative of denominator}): 3\cos x + 4\sin x = A(2\cos x + 5\sin x) + B(-2\sin x + 5\cos x). Comparing coefficients, 2A + 5B = 3 (for \cos x) and 5A - 2B = 4 (for \sin x). Solving this system yields A = \frac{26}{29} and B = \frac{7}{29}. The integral becomes \int(A + B\frac{f'(x)}{f(x)})dx = Ax + B\ln|f(x)|. Thus, \frac{\alpha}{29} = A = \frac{26}{29} \implies \alpha = 26.
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