To evaluate the integral \int \frac{{x-5}}{{(x-4)(x+7)2}}dx, we apply the method of partial fractions to express the integrand as a sum of simple rational functions of x:

\frac{{x-5}}{{(x-4)(x+7)2}}=A.

Which of the following options correctly completes the expression for A?

A. \frac{{1}}{{x-4}}+\frac{{B}}{{x+7}}+\frac{{C}}{{(x+7)2}}
B. \frac{{1}}{{x-4}}+\frac{{B}}{{(x+7)2}}+\frac{{C}}{{x+7}}
C. \frac{{1}}{{x+7}}+\frac{{B}}{{x-4}}+\frac{{C}}{{(x+7)2}}
D. \frac{{1}}{{(x+7)2}}+\frac{{B}}{{x-4}}+\frac{{C}}{{x+7}}