The logarithmic f′/f pattern

Material by the IntegralPaso project. Conditions and limits are stated in each guide; external teaching review is pending.

Look for the denominator derivative in the numerator, possibly with a constant factor. Work on intervals where f is nonzero.

Step-by-step example

1. Identify f and f′.

   $$\int\frac{2x}{x^2-4}\,dx$$

2. Make the complete substitution.

   $$u=x^2-4,\quad du=2x\,dx$$

3. Integrate and exclude x = ±2.

   $$I=\ln|x^2-4|+C$$