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

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

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Solution Steps

Step 1: Partial Fraction Decomposition

We start with the integral

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

Using partial fraction decomposition, we express the integrand as

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

Step 2: Integration

Next, we integrate each term separately:

(1x+5+4x+2)dx=1x+5dx+4x+2dx. \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:

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

where CC is the constant of integration.

Final Answer

Thus, the final result of the integral is

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

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