3.5.3—Partial fractions

After factoring the denominator, decompose a proper rational function into linear terms and irreducible quadratic terms. Integrate each using logarithm or arctangent forms.

Complete the square in a quadratic denominator before integrating. Repeated factors need one numerator for each power.

∫1/(x²+1)dx=tan⁻¹x+C, while ∫(2x)/(x²+1)dx=ln(x²+1)+C.

A quadratic denominator does not always produce a logarithm; if its numerator is constant after completing the square, an inverse tangent may appear.