To solve the integral

$\int \frac{(3\cos x - 2)\sin x}{5 - \sin^2 x - 4\cos x} \, dx,$

we can approach it using substitution. Let's define:

$u = 5 - \sin^2 x - 4\cos x.$

First, we need to find the derivative of $u$ with respect to $x$:

$\frac{du}{dx} = -2\sin x \cos x + 4\sin x.$

Notice that this can be factored as:

$\frac{du}{dx} = \sin x(-2\cos x + 4).$

This closely resembles the numerator of the integrand. However, the numerator is $(3\cos x - 2)\sin x$, so let's rewrite the integral in terms of $du$:

$\int \frac{(3\cos x - 2)\sin x}{5 - \sin^2 x - 4\cos x} \, dx.$

Using the substitution $u = 5 - \sin^2 x - 4\cos x$, the integral becomes:

$\int \frac{-(3\cos x - 2)}{-2\cos x + 4} \frac{du}{\sin x(-2\cos x + 4)}.$

Simplifying further:

$\int \frac{du}{u}.$

This integral is straightforward:

$\ln |u| + C,$

where $u = 5 - \sin^2 x - 4\cos x$. So the final answer is:

$\ln |5 - \sin^2 x - 4\cos x| + C.$

Would you like more details or have any questions?

Here are some related questions:

1. How do you perform trigonometric substitutions in integrals?
2. What are common strategies for solving integrals involving trigonometric functions?
3. How do you recognize a function that is suitable for $u$-substitution?
4. Can you generalize the integral approach for functions involving both sine and cosine?
5. How does the chain rule assist in evaluating integrals by substitution?

Tip: When dealing with integrals involving trigonometric functions, look for patterns or identities that can simplify the integrand before integrating.