Partial fractionsEdexcel International A Level Maths: Revision notes

A rational function is a ratio of two polynomials. Writing a rational function as a sum of simpler fractions, its partial fractions, makes it much easier to integrate, differentiate or expand as a series. A fraction is proper if the degree of the numerator is less than the degree of the denominator, and improper if it is equal or greater. This course covers denominators made of linear factors, with or without a repeated factor, such as $(ax+b)(cx+d)(ex+f)$ and $(ax+b)(cx+d)^2$. Quadratic factors such as $x^2+a$ are not required.

For a proper fraction with distinct linear factors, write $\frac{px+q}{(x-a)(x-b)}=\frac{A}{x-a}+\frac{B}{x-b}$. Multiply through by the denominator: $px+q=A(x-b)+B(x-a)$. This is an identity, true for every $x$, so choose convenient values: $x=a$ removes $B$ and $x=b$ removes $A$. Example: $\frac{10x-1}{(x-1)(2x+1)}=\frac{A}{x-1}+\frac{B}{2x+1}$ gives $10x-1=A(2x+1)+B(x-1)$. At $x=1$: $9=3A$, $A=3$. At $x=-\frac12$: $-6=-\frac32B$, $B=4$. So the fraction is $\frac{3}{x-1}+\frac{4}{2x+1}$. Check at $x=0$: the original gives $\frac{-1}{(-1)(1)}=1$ and the partial fractions give $\frac{3}{-1}+\frac41=1$.

A repeated factor $(x-a)^2$ needs two terms: $\frac{B}{x-a}+\frac{C}{(x-a)^2}$. For $\frac{3x^2+6}{(x+2)(x-1)^2}=\frac{A}{x+2}+\frac{B}{x-1}+\frac{C}{(x-1)^2}$ multiply out: $3x^2+6=A(x-1)^2+B(x+2)(x-1)+C(x+2)$. $x=-2$ gives $18=9A$, $A=2$. $x=1$ gives $9=3C$, $C=3$. Then compare coefficients of $x^2$: $3=A+B$, so $B=1$. (Or use $x=0$: $6=A-2B+2C$.) The answer is $\frac{2}{x+2}+\frac{1}{x-1}+\frac{3}{(x-1)^2}$.

If the degree of the numerator is equal to or greater than the degree of the denominator, first divide to obtain a polynomial plus a proper fraction. For $\frac{3x^2+6x-3}{(x-1)(x+2)}$ the denominator is $x^2+x-2$. Since $3x^2+6x-3=3(x^2+x-2)+3x+3$, the fraction is $3+\frac{3x+3}{(x-1)(x+2)}$, and then $\frac{3x+3}{(x-1)(x+2)}=\frac{2}{x-1}+\frac{1}{x+2}$. So the result is $3+\frac{2}{x-1}+\frac{1}{x+2}$. You can also include the constant straight away: write $P+\frac{Q}{x-1}+\frac{R}{x+2}$ and find $P$ first by comparing the $x^2$ coefficients.

Integration: $\int\frac{A}{ax+b}\,\mathrm{d}x=\frac{A}{a}\ln|ax+b|+c$ and $\int\frac{C}{(ax+b)^2}\,\mathrm{d}x=-\frac{C}{a(ax+b)}+c$. For example $\int_2^3h(x)\,\mathrm{d}x=\left[3x+2\ln|x-1|+\ln|x+2|\right]_2^3=3+\ln5$. Differentiation: write the function as a sum of terms such as $A(ax+b)^{-1}$ and differentiate each: $\frac{\mathrm{d}}{\mathrm{d}x}\frac{A}{ax+b}=-\frac{Aa}{(ax+b)^2}$. Series: expand each partial fraction with the binomial expansion and add. $\frac{5+x}{(1-x)(1+2x)}=\frac{2}{1-x}+\frac{3}{1+2x}=2(1+x+x^2+\ldots)+3(1-2x+4x^2-\ldots)=5-4x+14x^2+\ldots$, valid for $|x|<\frac12$.