What is the value of A ?
For the next two (02) items that follow : Let \( \int \frac{\sin \theta d\theta}{(2 + \cos \theta)(3 + 4\cos \theta)} = A \ln |2 + \cos \theta| + B \ln |3 + 4\cos \theta| \)
- A. -\frac{2}{5}
- B. -\frac{1}{5}
- C. \frac{1}{5} ✓
- D. \frac{2}{5}
Correct Answer: C. \frac{1}{5}
Explanation
Substitute \( \cos \theta = t \), so \( -\sin \theta d\theta = dt \). The integral becomes \( \int \frac{-dt}{(t+2)(4t+3)} \). Using partial fractions, \( \frac{-1}{(t+2)(4t+3)} = \frac{1/5}{t+2} - \frac{4/5}{4t+3} \). Integrating gives \( \frac{1}{5} \ln|t+2| - \frac{1}{5} \ln|4t+3| \). Comparing coefficients, A = 1/5.
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