Evaluate $\int_{-7}^9\frac{-4}{x}dx$

Question

5.4

Can somebody verify this solution for me?? Thanks!!

Evaluate $\int_{-7}^9\frac{-4}{x}dx$

---

$\int_{-7}^9\frac{-4}{x}dx$

$=-4\int_{-7}^9\frac{1}{x}dx$

$=-4ln(|x|)|_{-7}^9dx$

$=-4(ln(|9|)-ln(|-7|))$

$=-4(ln(9)-ln(7))$

$=-4ln(\frac{9}{7})$

Answer 1

The function $-\frac{4}{x}$ has a singularity near $x = 0$. You have to deal with it somehow, e.g.

$$ \begin{aligned} -\text{P.V.}\int_{-7}^9\frac{4}{x}dx &= \lim_{\varepsilon \to 0}\left(-4\left(\int_{-7}^{-\varepsilon}\frac{1}{x}dx + \int_{+\varepsilon}^9\frac{1}{x}dx\right)\right) \end{aligned} $$

Answer 2

$\int^9_{-7}{\frac{-4}{x}dx}=\int^0_{-7}{\frac{-4}{x}dx}+\int^9_{0}{\frac{-4}{x}dx} = \lim_{a \rightarrow 0^-}\int^a_{-7}{\frac{-4}{x}dx} + lim_{b \rightarrow 0^+}\int^9_{b}{\frac{-4}{x}dx}$

$=-4\lim_{a \rightarrow 0^-}(ln|a|-ln|-7|) -4\lim_{b \rightarrow 0^+}(ln|9|-ln|b|) \quad $ integral does not converge!