Exercise 15

In Table in Activity, we noted that for $x \gt 0$, an antiderivative of $f(x) = \frac{1}{x}$ is $F(x) = \ln(x)$. Here we observe that a key difference between $f(x)$ and $F(x)$ is that $f$ is defined for all $x \ne 0$, while $F$ is only defined for $x \gt 0$, and see how we can actually define an antiderivative of $f$ for all values of $x$.

1. Suppose that $x \lt 0$, and let $G(x) = \ln(-x)$. Compute $G'(x)$.

2. Explain why $G$ is an antiderivative of $f$ for $x \lt 0$.

3. Let $H(x) = \ln(|x|)$, and recall that $$\begin{aligned} |x| = \begin{cases} -x, & \ \text{if} \ x \lt 0 \\ x, & \ \text{if} \ x \ge 0 \end{cases} \end{aligned}$$. Explain why $H(x) = G(x)$ for $x \lt 0$ and $H(x) = F(x)$ for $x \gt 0$.

4. Now discuss why we say that an antiderivative of $\frac{1}{x}$ is $\ln(|x|)$ for all $x \ne 0$.