Partial fractions in integration

Integration by partial fractions is a technique to facilitate the integration of a rational expression by partial fraction decomposition.

Given an integral
$\int {\frac {f(x)}{g(x)}}dx$
where $f(x)$ and $g(x)$ are both polynomials, integration by partial fractions shows how to separate the problem into multiple integrals before integrating.

A 1st-Degree Denominator

These are a few methods of solving integrals with first degree denominators.

A 1st-Degree Denominator

Given:
$\int {\frac {1}{ax+b}}dx$
Substitute $u=ax+b$
$=\int {\frac {1}{u}}{\frac {du}{a}}={\frac {1}{a}}\int {\frac {du}{u}}={\frac {1}{a}}\ln {|u|}+C={\frac {1}{a}}\ln {|ax+b|}+C$

This means that if given an integral such as:
$\int {\frac {8}{3x+13}}dx$
The steps can be skipped by using the general formula above get:
$={\frac {8}{3}}\ln {|3x+13|}+C$

A Repeated 1st-Degree Denominator

The formula for integrals where a first degree polynomial denominator is raised to a power greater than one is much different than the formula above. Given
$\int {\frac {1}{(ax+b)^{k}}}dx$
$u=ax+b$
$=\int {\frac {1}{u^{k}}}{\frac {du}{a}}={\frac {1}{a}}\int u^{-k}du={\frac {1}{a}}\cdot {\frac {u^{k-1}}{-k}}+C={-1 \over kau^{k-1}}+C={-1 \over {ka(ax+b)^{k-1}}}+C$
This means that integrals such 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 \int {5 \over {(3x+17)^{12}}}dx}
are now very easy:
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 = {-5 \over {(3)(12)(3x+17)^{11}}}+C = {-5 \over {36(3x+17)^{11}}}+C}

A 2nd-Degree Denominator

2nd-Degree Denominators get more complicated, especially with those that do not factor.

A Reducible 2nd-Degree Polynomial 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 \int\frac{3x+11}{x^2-x-6}dx}

The first step is to factor the denominator as much as possible and get the form of the partial fraction decomposition. Doing this 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 \frac{3x+11}{(x-3)(x+2)}=\frac{A}{x-3}+\frac{B}{x+2}}

This allows the denominator to split the fraction in to sums by cross multiplying the denominators,

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{3x+11}{(x-3)(x+2)}=\frac{A(x+2)+B(x-3)}{(x-3)(x+2)}}

Therefore the problem can be restated,

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 3x+11=A(x+2)+B(x-3)}

Now it is possible to solve for A and B by substituting x with a value that allows the term to go to 0. For example,

Let :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 x=-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 3(-2)+11=A(-2+2)+B(-2-3)}

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 5=A(0)+B(5)}

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=-1}

Let :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 x=3} ,

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 3(3)+11=A(3+2)+B(3-3)}

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 20=A(5)+B(0)}

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=4}

Then plug in the values of A and B and 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 \frac{4}{x-3}-\frac{1}{x+2}}

Now solve the 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+11}{x^2-x-6}dx=\int\frac{4}{x-3}-\frac{1}{x+2}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{3x+11}{x^2-x-6}dx=\int\frac{4}{x-3}dx-\int\frac{1}{x+2}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{3x+11}{x^2-x-6}dx=4ln|x-3|-ln|x+2|+c}