Solve ∫ 1/x dx = ln|x| + C

Step-by-step solution

1. The power rule for integration $\int x^n \, dx = \frac{x^{n+1}}{n+1}$ fails when $n = -1$ (would divide by zero).

2. Use the special antiderivative: $\frac{d}{dx}\ln|x| = \frac{1}{x}$.

3. Therefore $\int \frac{1}{x} \, dx = \ln|x| + C$.

4. The absolute value ensures the result is valid for negative $x$ too (where $\ln(x)$ would be undefined in reals).