Partial Fractions

Suppose we want to find Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\frac{3x+1}{x^2+x}dx} . One way to do this is to simplify the integrand by finding constants $A$ and $B$ so that

${\frac {3x+1}{x^{2}+x}}={\frac {3x+1}{x(x+1)}}={\frac {A}{x}}+{\frac {B}{x+1}}$ .

This can be done by cross multiplying the fraction which gives

${\frac {3x+1}{x(x+1)}}={\frac {A(x+1)+Bx}{x(x+1)}}$

As both sides have the same denominator we must have

$3x+1=A(x+1)+Bx$

This is an equation for $x$ so it must hold whatever value $x$ is. If we put in $x=0$ we get Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A=1} and putting $x=-1$ gives Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =-B=-2} so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B=2} . So we see that

${\frac {3x+1}{x^{2}+x}}={\frac {1}{x}}+{\frac {2}{x+1}}$

Returning to the original integral

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\frac{3x+1}{x^2+x}dx}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\int\frac{dx}{x}+\int\frac{2}{x+1}dx}
	$=\int {\frac {dx}{x}}+2\int {\frac {dx}{x+1}}$
	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\ln|x|+2\ln\Big|x+1\Big|+C}

Rewriting the integrand as a sum of simpler fractions has allowed us to reduce the initial integral to a sum of simpler integrals. In fact this method works to integrate any rational function.

Method of Partial Fractions

To decompose the rational function ${\frac {P(x)}{Q(x)}}$:

• Step 1 Use long division (if necessary) to ensure that the degree of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(x)} is less than the degree of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q(x)} .
• Step 2 Factor Q(x) as far as possible.
• Step 3 Write down the correct form for the partial fraction decomposition (see below) and solve for the constants.

To factor Q(x) we have to write it as a product of linear factors (of the form Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ax+b} ) and irreducible quadratic factors (of the form Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ax^2+bx+c} with $b^{2}-4ac<0$).

Some of the factors could be repeated. For instance if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q(x) = x^3-6x^2+9x} we factor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q(x)} as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q(x)=x(x^2-6x+9)=x(x-3)(x-3)=x(x-3)^2}

It is important that in each quadratic factor we have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b^2-4ac<0} , otherwise it is possible to factor that quadratic piece further. For example if $Q(x)=x^{3}-3x^{2}+2x$ then we can write

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q(x)=x(x^2-3x+2)=x(x-1)(x-2)}

We will now show how to write Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{P(x)}{Q(x)}} as a sum of terms of the form

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{A}{(ax+b)^k}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{Ax+B}{(ax^2+bx+c)^k}}

Exactly how to do this depends on the factorization of $Q(x)$ and we now give four cases that can occur.

Q(x) is a product of linear factors with no repeats

This means that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q(x)=(a_1x+b_1)(a_2x+b_2)\cdots(a_nx+b_n)} where no factor is repeated and no factor is a multiple of another.

For each linear term we write down something of the form Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{A}{(ax+b)}} , so in total we write

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{P(x)}{Q(x)}=\frac{A_1}{a_1x+b_1}+\frac{A_2}{a_2x+b_2}+\cdots+\frac{A_n}{a_nx+b_n}}

Example Problem

Find Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\frac{1+x^2}{(x+3)(x+5)(x+7)}dx}

Here we have $P(x)=1+x^{2}\ ,\ Q(x)=(x+3)(x+5)(x+7)$ and Q(x) is a product of linear factors. So we write

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1+x^2}{(x+3)(x+5)(x+7)}=\frac{A}{x+3}+\frac{B}{x+5}+\frac{C}{x+7}}

Multiply both sides by the denominator

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1+x^2=A(x+5)(x+7)+B(x+3)(x+7)+C(x+3)(x+5)}

Substitute in three values of x to get three equations for the unknown constants,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{matrix} x=-3 & 1+3^2=2\cdot 4 A \\ x=-5 & 1+5^2=-2\cdot 2 B \\ x=-7 & 1+7^2=(-4)\cdot (-2)C \end{matrix}}

so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A=\tfrac{5}{4}\ ,\ B=-\tfrac{13}{2}\ ,\ C=\tfrac{25}{4}} , and

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1+x^2}{(x+3)(x+5)(x+7)}=\frac{5}{4x+12}-\frac{13}{2x+10}+\frac{25}{4x+28}}

We can now integrate the left hand side.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\frac{1+x^2}{(x+3)(x+5)(x+7)}dx=\tfrac{5}{4}\ln\Big|x+3\Big|-\tfrac{13}{2}\ln\Big|x+5\Big|+\tfrac{25}{4}\ln\Big|x+7\Big|+C}

Exercises

Evaluate the following by the method partial fraction decomposition.

1. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\frac{2x+11}{(x+6)(x+5)}dx}

2. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\frac{7x^2-5x+6}{(x-1)(x-3)(x-7)}dx}

Q(x) is a product of linear factors some of which are repeated

If $(ax+b)$ appears in the factorisation of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q(x)} k-times then instead of writing the piece Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{A}{ax+b}} we use the more complicated expression

${\frac {A_{1}}{ax+b}}+{\frac {A_{2}}{(ax+b)^{2}}}+{\frac {A_{3}}{(ax+b)^{3}}}+\cdots +{\frac {A_{k}}{(ax+b)^{k}}}$

Example 2

Find $\int {\frac {dx}{(x+1)(x+2)^{2}}}$

Here $P(x)=1$ and $Q(x)=(x+1)(x+2)^{2}$ We write

${\frac {1}{(x+1)(x+2)^{2}}}={\frac {A}{x+1}}+{\frac {B}{x+2}}+{\frac {C}{(x+2)^{2}}}$

Multiply both sides by the denominator $1=A(x+2)^{2}+B(x+1)(x+2)+C(x+1)$

Substitute in three values of $x$ to get 3 equations for the unknown constants,

${\begin{matrix}x=0&1=2^{2}A+2B+C\\x=-1&1=A\\x=-2&1=-C\end{matrix}}$

so $A=1\ ,\ B=-1\ ,\ C=-1$ and

${\frac {1}{(x+1)(x+2)^{2}}}={\frac {1}{x+1}}-{\frac {1}{x+2}}-{\frac {1}{(x+2)^{2}}}$

We can now integrate the left hand side.

$\int {\frac {dx}{(x+1)(x+2)^{2}}}=\ln \left|x+1\right|-\ln \left|x+2\right|+{\frac {1}{x+2}}+C$

We now simplify the fuction with the property of Logarithms.

$\ln \left|x+1\right|-\ln \left|x+2\right|+{\frac {1}{x+2}}+C=\ln \left|{\frac {x+1}{x+2}}\right|+{\frac {1}{x+2}}+C$

Exercise

3. Evaluate $\int {\frac {x^{2}-x+2}{x(x+2)^{2}}}dx$ using the method of partial fractions.

Q(x) contains some quadratic pieces which are not repeated

If $ax^{2}+bx+c$ appears we use ${\frac {Ax+B}{ax^{2}+bx+c}}$ .

Exercises

Evaluate the following using the method of partial fractions.

4. $\int {\frac {2}{(x+2)(x^{2}+3)}}dx$

5. $\int {\frac {dx}{(x+2)(x^{2}+2)}}$

Q(x) contains some repeated quadratic factors

If $ax^{2}+bx+c$ appears k-times then use

${\frac {A_{1}x+B_{1}}{ax^{2}+bx+c}}+{\frac {A_{2}x+B_{2}}{(ax^{2}+bx+c)^{2}}}+{\frac {A_{3}x+B_{3}}{(ax^{2}+bx+c)^{3}}}+\cdots +{\frac {A_{k}x+B_{k}}{(ax^{2}+bx+c)^{k}}}$

Exercise

Evaluate the following using the method of partial fractions.

6. $\int {\frac {dx}{(x-1)(x^{2}+1)^{2}}}$

Exercies Solutions

1. $\ln {\Big |}x+6{\Big |}+\ln {\Big |}x+5{\Big |}+C$
2. ${\tfrac {2}{3}}\ln {\Big |}x-1{\Big |}-{\tfrac {27}{4}}\ln {\Big |}x-3{\Big |}+{\tfrac {157}{12}}\ln {\Big |}x-7{\Big |}+C$
3. ${\frac {\ln {\Big |}x(x+2){\Big |}}{2}}+{\frac {4}{x+2}}+C$
4. $\ln \left({\sqrt[{7}]{\frac {(x+2)^{2}}{x^{2}+3}}}\right)+{\frac {4\arctan {\Big (}{\tfrac {x}{\sqrt {3}}}{\Big )}}{7{\sqrt {3}}}}+C$
5. $\ln \left({\sqrt[{12}]{\frac {(x+2)^{2}}{x^{2}+2}}}\right)+{\frac {{\sqrt {2}}\arctan {\Big (}{\tfrac {x}{\sqrt {2}}}{\Big )}}{6}}+C$
6. ${\frac {1-x}{4(x^{2}+1)}}+{\tfrac {1}{8}}\ln \left({\frac {(x-1)^{2}}{x^{2}+1}}\right)-{\frac {\arctan(x)}{2}}+C$

Resources

Integration Using Partial Fraction Decomposition Part 1 by James Sousa

Integration Using Partial Fraction Decomposition Part 2 by James Sousa

Partial Fraction Decomposition Part 1 (Linear) by James Sousa

Partial Fraction Decomposition - Part 2 of 2 by James Sousa

Partial Fraction Decomposition - Example 1 by patrickJMT

Partial Fraction Decomposition - Example 2 by patrickJMT

Partial Fraction Decomposition - Example 4 by patrickJMT

Partial Fraction Decomposition - Example 5 by patrickJMT

Partial Fraction Decomposition - Example 6 by patrickJMT

Partial Fraction Decompositions by patrickJMT