Proper fraction: lower numerator degree

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

Compare polynomial degrees, not their values. A lower numerator degree means no preliminary division is needed. Factor the denominator, look for a direct substitution and then decide whether partial fractions help.

Step-by-step example

1. The numerator has degree 1 and the denominator degree 2.

   $$\int\frac{x}{x^2+1}\,dx$$

2. Recognize the denominator derivative and substitute.

   $$u=x^2+1,\quad du=2x\,dx$$

3. Integrate and return to the original variable.

   $$I=\frac12\ln(x^2+1)+C$$