All courses › Mathematics R2 › Partial fractions

Partial fractions

A fraction with a factorised denominator can be written as a sum of simpler fractions. For two linear factors with numerator 1 the coefficients are 1 and $-1$ . You check by using a common denominator, or by inserting a number where the denominator is not zero. The integral of each partial fraction is a logarithm. The method is used when substitution and integration by parts do not fit.

\frac{1}{x(x+1)} = \frac{1}{x} - \frac{1}{x+1}

partial fractions

\int\frac{1}{x(x+1)}\,dx = \ln|x| - \ln|x+1| + C

the integral

Symbols

x

the variable

Example

At $x = 2$ , $\dfrac{1}{2\cdot 3} = \dfrac{1}{6}$ and $\dfrac{1}{2} - \dfrac{1}{3} = \dfrac{1}{6} \approx 0.1667$ .

Check the splitting with a number before you integrate. The common denominator should give back the original fraction.

Practise integration techniques for free →

← Integration by parts of xe^x · Area between graphs →

Part of Mathematics R2: Integration techniques.