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

Transcript text: Evaluate $\int \frac{5 x+22}{(x+5)(x+2)} d x$

Solution

Step 1: Partial Fraction Decomposition

We start with the integral

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

Using partial fraction decomposition, we express the integrand as

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

Step 2: Integration

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.

Thus, the final result of the integral is

\[ \boxed{\log |x + 5| + 4 \log |x + 2| + C}. \]

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