\( \frac { ( n + 1 ) ! } { ( n + 1 ) ^ { n + 1 } } \leq \frac { 1 } { 2 ^ { n } } \)

8.- Resolver por Integración por fracciones parciales \( \int \frac{1}{x^{2}-4} d x \) Opciones: a) \( \frac{1}{2} \ln |x-2|-\frac{1}{2} \ln |x+2|+C \) b) ln \( |x-2|-\ln |x+2|+C \) c) \( \frac{1}{2} \ln |x+2|-\frac{1}{2} \ln |x-2|+C \) d) \( \left.{ }^{\frac{1}{2}} \ln \left|x+2+\frac{1}{2} \ln \right| x-2 \right\rvert\,+C \)

Find the difference quotient of \( f \); that is, find \( \frac{f(x+h)-f(x)}{h}, h \neq 0 \), for the following function. Be sure to simplify. \( f(x)=3 x^{2}-7 x+8 \) \( \frac{f(x+h)-f(x)}{h}=\square \)

Suppose that the functions \( u \) and \( w \) are defined as follows. \( \begin{array}{l}u(x)=-5 x+1 \\ w(x)=-x+5\end{array} \) Find the following. \( \begin{array}{l}(w \circ u)(1)=\square \\ (u \circ w)(1)=\square\end{array} \)

\( \int _{}^{} d m \)

If \( f(x)=\frac{7}{x+5} \) and \( g(x)=\frac{x+73}{x^{2}-25} \), determine \( (f+g)(x) \) \( (f+g)(x)=\frac{\pi+32}{x^{2}-25} \) \( (f+g)(x)=\frac{x+32}{x+5} \) \( (f+g)(x)=\frac{8}{x^{2}-25} \) \( (f+g)(x)=\frac{8}{x-5} \)