Step-by-Step Solution
Key Concept: Express the numerator as a linear combination of the denominator and its derivative to separate the integral into a polynomial part and a logarithmic part.
We express the numerator as $5\sin x = a(\sin x - 2\cos x) + b(\cos x + 2\sin x)$. Expanding and comparing coefficients: $5\sin x = a\sin x - 2a\cos x + b\cos x + 2b\sin x$. This gives $a + 2b = 5$ and $-2a + b = 0$, so $b = 2a$, yielding $a + 4a = 5$, thus $a = 1$ and $b = 2$. Therefore $\int \frac{5\sin x}{\sin x - 2\cos x} dx = \int 1 dx + 2\int \frac{\cos x + 2\sin x}{\sin x - 2\cos x} dx = x + 2\ln|\sin x - 2\cos x| + k$. The value of $a$ is $2$.
Correct Answer: 2