Integrating with Partial Fractions

Split the fraction before integrating

Partial fractions express one rational expression as a sum or difference of simpler fractions. Factor the denominator first, then choose the required partial-fraction terms and determine their coefficients.

1. Factor the denominator. Write a fraction for each required linear factor or power of a repeated factor.
2. Find the coefficients. Multiply through by the original denominator, then substitute convenient values of $x$ to determine the unknowns.
3. Integrate the simpler terms. Allow for the coefficient of $x$ in each denominator, simplify and include $+C$ for an indefinite integral.

Linear denominators give logarithms

For constants $A$, $a$ and $b$, with $a\ne0$:

The factor $1/a$ compensates for the derivative of $ax+b$. Keep the modulus signs in the logarithm.

Worked example: two distinct factors

Find $\displaystyle I=\int\frac{3}{x^2+5x+4}\,dx$. Since $x^2+5x+4=(x+1)(x+4)$, set

$$\frac{3}{(x+1)(x+4)}=\frac{A}{x+1}+\frac{B}{x+4}.$$

Multiplying by $(x+1)(x+4)$ gives the polynomial identity $3=A(x+4)+B(x+1)$. Choose values that remove one unknown at a time:

$$\begin{aligned} x=-1:&\quad 3=3A &&\Rightarrow A=1,\\ x=-4:&\quad 3=-3B &&\Rightarrow B=-1. \end{aligned}$$

Now integrate the two partial fractions:

$$\begin{aligned} I&=\int\left(\frac{1}{x+1}-\frac{1}{x+4}\right)\,dx\\ &=\ln|x+1|-\ln|x+4|+C. \end{aligned}$$