Questions: Evaluate ∫ (5x+22)/(x+5)(x+2) dx

Solution Steps

We start with the integral

$$\int \frac{5x + 22}{(x + 5)(x + 2)} \, dx.$$

Using partial fraction decomposition, we express the integrand as

$$\frac{5x + 22}{(x + 5)(x + 2)} = \frac{1}{x + 5} + \frac{4}{x + 2}.$$

Next, we integrate each term separately:

$$\int \left( \frac{1}{x + 5} + \frac{4}{x + 2} \right) \, dx = \int \frac{1}{x + 5} \, dx + \int \frac{4}{x + 2} \, dx.$$

The integrals yield:

$$\log |x + 5| + 4 \log |x + 2| + C,$$

where $C$ is the constant of integration.

Final Answer