Integrating with partial fractions

Splitting a rational function into partial fractions so each piece integrates to a logarithm — turning an impossible-looking fraction into a sum of ln terms.

Pure · Integration

Integrating with partial fractions

Splitting a rational function into partial fractions so each piece integrates to a logarithm — turning an impossible-looking fraction into a sum of ln terms.

Why it works

A fraction like

\dfrac{3x + 5}{(x + 1)(x + 3)}

has no standard integral as it stands — the bottom isn't the derivative of anything tidy, so the reverse chain rule won't bite. But you already know how to break such a fraction into partial fractions, and each piece is something you can integrate.

The key fact is that a single partial fraction integrates straight to a logarithm:

\int \frac{A}{ax + b}\,dx = \frac{A}{a}\ln|ax + b| + c.

So once a rational function is written as a sum of terms

\dfrac{A}{x - p} + \dfrac{B}{x - q} + \dots

, its integral is just a sum of logs. The method is always the same: decompose into partial fractions, then integrate each term to a logarithm.

Two things to watch:

The fraction must be proper (numerator degree below denominator degree) before

decomposing. If it isn't, do polynomial division first — the quotient integrates normally and the proper remainder splits into partial fractions.

A repeated factor $(x - p)^2$ contributes a term $\dfrac{C}{(x-p)^2}$ , which

integrates by the power rule (to

-\dfrac{C}{x - p}

), not to a log. Only the single-power factors give logarithms.

A definite integral then collects the logs and tidies them with the laws of logs — often into a single clean

\ln

of a fraction.