We start with the integral
∫5x+22(x+5)(x+2) dx. \int \frac{5x + 22}{(x + 5)(x + 2)} \, dx. ∫(x+5)(x+2)5x+22dx.
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}. (x+5)(x+2)5x+22=x+51+x+24.
Next, we integrate each term separately:
∫(1x+5+4x+2) dx=∫1x+5 dx+∫4x+2 dx. \int \left( \frac{1}{x + 5} + \frac{4}{x + 2} \right) \, dx = \int \frac{1}{x + 5} \, dx + \int \frac{4}{x + 2} \, dx. ∫(x+51+x+24)dx=∫x+51dx+∫x+24dx.
The integrals yield:
log∣x+5∣+4log∣x+2∣+C, \log |x + 5| + 4 \log |x + 2| + C, log∣x+5∣+4log∣x+2∣+C,
where CCC is the constant of integration.
Thus, the final result of the integral is
log∣x+5∣+4log∣x+2∣+C. \boxed{\log |x + 5| + 4 \log |x + 2| + C}. log∣x+5∣+4log∣x+2∣+C.
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