Learning Outcomes
- I can integrate functions using the method of partial fractions.
Section 5.6 Partial fractions (TI6)
Subsection 5.6.1 Activities
Activity 5.6.1.
Consider \(\displaystyle \int \frac{x^2+x+1}{x^3+x} \,dx\text{.}\) Which substitution would you choose to evaluate this integral?
- \(\displaystyle u=x^3\)
- \(\displaystyle u=x^3+x\)
- \(\displaystyle u=x^2+x+1\)
- Substitution is not effective
Activity 5.6.2.
Using the method of substitution, which of these is equal to \(\displaystyle\int \frac{5}{x+7} dx\text{?}\)
- \(\displaystyle 5\ln|x+7| +C\)
- \(\displaystyle \frac{5}{7}\ln|x+7| +C\)
- \(\displaystyle 5\ln|x|+5\ln|7|+C\)
- \(\displaystyle \frac{5}{7}\ln|x|+C\)
Observation 5.6.3.
To avoid repetitive substitution, the following integral formulas will be useful.
\begin{equation*}
\int\frac{1}{x+b}dx=\ln|x+b|+C
\end{equation*}
\begin{equation*}
\int\frac{1}{(x+b)^2}dx=-\frac{1}{x+b}+C
\end{equation*}
\begin{equation*}
\int\frac{1}{x^2+b^2}dx=\frac{1}{b}\arctan\left(\frac{x}{b}\right)+C
\end{equation*}
Activity 5.6.4.
Which of the following is equal to \(\displaystyle\frac{1}{x}+\frac{1}{x^2+1}\text{?}\)
- \(\displaystyle \frac{2x}{x^2+x+1}\)
- \(\displaystyle \frac{x^3+x}{x^2+x+1}\)
- \(\displaystyle \frac{2x}{x^3+x}\)
- \(\displaystyle \frac{x^2+x+1}{x^3+x}\)
Activity 5.6.5.
Based on the previous activities, which of these is equal to \(\displaystyle\int \frac{x^2+x+1}{x^3+x} dx\text{?}\)
- \(\displaystyle \ln|x|+\arctan(x)+C\)
- \(\displaystyle \ln|x^2+x+1|+C\)
- \(\displaystyle \ln|x^3+x|+C\)
- \(\displaystyle \arctan(x^3+x)+C\)
Activity 5.6.6.
Suppose we know
\begin{equation*}
\frac{10x+11}{x^2+x-2}=\frac{7}{x-1}+\frac{3}{x+2}\text{.}
\end{equation*}
Which of these is equal to \(\int\frac{10x+11}{x^2+x-2} dx\text{?}\)
- \(\displaystyle 7\ln|x-1|+3\arctan(x+2)+C\)
- \(\displaystyle 7\ln|x-1|+3\ln|x+2|+C\)
- \(\displaystyle 7\arctan(x-1)+3\arctan(x+2)+C\)
- \(\displaystyle 7\arctan(x-1)+3\ln|x+2|+C\)
Observation 5.6.7.
To find integrals like \(\int \frac{x^2+x+1}{x^3+x} dx\) and \(\int\frac{10x+11}{x^2+x-2} dx\text{,}\) we’d like to decompose the fractions into simpler partial fractions that may be integrated with these formulas
\begin{equation*}
\int\frac{1}{x+b}dx=\ln|x+b|+C
\end{equation*}
\begin{equation*}
\int\frac{1}{(x+b)^2}dx=-\frac{1}{x+b}+C
\end{equation*}
\begin{equation*}
\int\frac{1}{x^2+b^2}dx=\frac{1}{b}\arctan\left(\frac{x}{b}\right)+C
\end{equation*}
Fact 5.6.8. Partial Fraction Decomposition.
Let \(\displaystyle \frac{p(x)}{q(x)}\) be a rational function, where the degree of \(p\) is less than the degree of \(q\text{.}\)
- Linear Terms: Let \((x-a)^n\) divide \(q(x)\text{,}\) Then the decomposition of \(\frac{p(x)}{q(x)}\) will contain the terms\begin{equation*} \frac{A_1}{(x-a)} + \frac{A_2}{(x-a)^2} + \cdots +\frac{A_n}{(x-a)^n}\text{.} \end{equation*}
- Quadratic Terms: Let \((x^2+bx+c)^n\) divide \(q(x)\text{,}\) where \(x^2+bx+c\) is irreducible. Then the decomposition of \(\frac{p(x)}{q(x)}\) will contain the terms\begin{equation*} \frac{B_1x+C_1}{x^2+bx+c}+\frac{B_2x+C_2}{(x^2+bx+c)^2}+\cdots+\frac{B_nx+C_n}{(x^2+bx+c)^n}\text{.} \end{equation*}
Example 5.6.9.
Following is an example of a rather involved partial fraction decomposition.
\begin{align*}
&\frac{7 \, x^{6} - 4 \, x^{5} + 41 \, x^{4} - 20 \, x^{3} + 24 \, x^{2} + 11 \, x + 16}{x(x-1)^2(x^2+4)^2}\\
=& \frac{A}{x}+\frac{B}{x-1}+\frac{C}{(x-1)^2}+\frac{Dx+E}{x^2+4}+\frac{Fx+G}{(x^2+4)^2}
\end{align*}
Using some algebra, it’s possible to find values for \(A\) through \(G\) to determine
\begin{align*}
&\frac{7 \, x^{6} - 4 \, x^{5} + 41 \, x^{4} - 20 \, x^{3} + 24 \, x^{2} + 11 \, x + 16}{x(x-1)^2(x^2+4)^2}\\
=& \frac{1}{x}+\frac{2}{x-1}+\frac{3}{(x-1)^2}+\frac{4x+5}{x^2+4}+\frac{6x+7}{(x^2+4)^2}\text{.}
\end{align*}
Activity 5.6.10.
Which of the following is the form of the partial fraction decomposition of \(\displaystyle\frac{x^3-7x^2-7x+15}{x^3(x+5)}\text{?}\)
- \(\displaystyle \frac{A}{x}+\frac{B}{x+5}\)
- \(\displaystyle \frac{A}{x^3}+\frac{B}{x+5}\)
- \(\displaystyle \frac{A}{x}+\frac{B}{x^2}+ \frac{C}{x^3}+\frac{D}{x+5}\)
- \(\displaystyle \frac{A}{x}+\frac{B}{x^2}+ \frac{C}{x^3}+\frac{Dx+E}{x+5}\)
Activity 5.6.11.
Which of the following is the form of the partial fraction decomposition of \(\displaystyle\frac{x^2+1}{(x-3)^2(x^2+4)^2}\text{?}\)
- \(\displaystyle \frac{A}{x-3}+\frac{B}{(x-3)^2}+\frac{C}{x^2+4}+\frac{D}{(x^2+4)^2}\)
- \(\displaystyle \frac{A}{x-3}+\frac{B}{(x-3)^2}+\frac{Cx+D}{(x^2+4)^2}\)
- \(\displaystyle \frac{A}{x-3}+\frac{B}{(x-3)^2}+\frac{C}{x^2+4}+\frac{Dx+E}{(x^2+4)^2}\)
- \(\displaystyle \frac{A}{x-3}+\frac{B}{(x-3)^2}+\frac{Cx+D}{x^2+4}+\frac{Ex+F}{(x^2+4)^2}\)
Activity 5.6.12.
Consider that the partial decomposition of \(\displaystyle \frac{x^2+5x+3}{(x+1)^2x}\) is
\begin{equation*}
\displaystyle \frac{x^2+5x+3}{(x+1)^2x}=\frac{A}{x+1}+\frac{B}{(x+1)^2}+\frac{C}{x}.
\end{equation*}
What equality do we obtain if we multiply both sides of the above equation by \((x+1)^2x\text{?}\)
- \(\displaystyle x^2+5x+3=Ax(x+1)+Bx+C(x+1)^2\)
- \(\displaystyle x^2+5x+3=A(x+1)+B(x+1)^2+Cx\)
- \(\displaystyle x^2+5x+3=Ax(x+1)+Bx+C(x+1)\)
- \(\displaystyle x^2+5x+3=Ax(x+1)+Bx^2+C(x+1)^2\)
Activity 5.6.13.
Use your choice in Activity 5.6.12 (which must hold for any \(x\) value) to answer the following.
(a)
By substituting \(x=0\) into the equation, we may find:
- \(\displaystyle A=1\)
- \(\displaystyle B=-2\)
- \(\displaystyle C=3\)
(b)
By substituting \(x=-1\) into the equation, we may find:
- \(\displaystyle A=-4\)
- \(\displaystyle B=1\)
- \(\displaystyle C=5\)
Activity 5.6.14.
Using the results of Activity 5.6.13, show how to rewrite our choice from Activity 5.6.12
\begin{equation*}
\unknown x^2+\unknown x=Ax^2+Ax\text{.}
\end{equation*}
What value of \(A\) satisfies this equation?
- \(\displaystyle -2\)
- \(\displaystyle 3\)
- \(\displaystyle 4\)
- \(\displaystyle -5\)
Activity 5.6.15.
By using the form of the decomposition \(\displaystyle \frac{x^2+5x+3}{(x+1)^2x}=\frac{A}{x+1}+\frac{B}{(x+1)^2}+\frac{C}{x}\) and the coefficients found in Activity 5.6.13 and Activity 5.6.14, evaluate \(\displaystyle \int \frac{x^2+5x+3}{(x+1)^2x} dx\text{.}\)
Activity 5.6.16.
Given that \(\displaystyle\frac{x^3-7x^2-7x+15}{x^3(x+5)}=\frac{A}{x}+\frac{B}{x^2}+ \frac{C}{x^3}+\frac{D}{x+5}\) do the following to find \(A, B, C\text{,}\) and \(D\text{.}\)
(a)
Eliminate the fractions to obtain
\begin{equation*}
x^3-7x^2-7x+15=A(\unknown)(\unknown)+B(\unknown)(\unknown)+C(\unknown)+D(\unknown)\text{.}
\end{equation*}
(b)
Plug in an \(x\) value that lets you find the value of \(C\text{.}\)
(c)
Plug in an \(x\) value that lets you find the value of \(D\text{.}\)
(d)
Use other algebra techniques to find the values of \(A\) and \(B\text{.}\)
Activity 5.6.17.
Given your choice in Activity 5.6.16 Find \(\displaystyle\int \frac{x^3-7x^2-7x+15}{x^3(x+5)} dx.\)
Activity 5.6.18.
Consider the rational expression \(\displaystyle\frac{2x^3+2x+4}{x^4+2x^3+4x^2}.\) Which of the following is the partial fraction decomposition of this rational expression?
- \(\displaystyle \frac{1}{x}+\frac{1}{x^2}+\frac{2x-1}{x^2+2x+4}\)
- \(\displaystyle \frac{2}{x}+\frac{0}{x^2}+\frac{-1}{x^2+2x+4}\)
- \(\displaystyle \frac{0}{x}+\frac{1}{x^2}+\frac{-1}{x^2+2x+4}\)
- \(\displaystyle \frac{0}{x}+\frac{1}{x^2}+\frac{2x-1}{x^2+2x+4}\)
Activity 5.6.19.
Given your choice in Activity 5.6.18 Find \(\displaystyle\int \frac{2x^3+2x+4}{x^4+2x^3+4x^2} dx\text{.}\)
Activity 5.6.20.
Given that \(\displaystyle \frac{2x+5}{x^2+3x+2}=\frac{-1}{x+2}+\frac{3}{x+1}\text{,}\) find \(\displaystyle\int_0^3 \frac{2x+5}{x^2+3x+2} dx\text{.}\)
Activity 5.6.21.
Evaluate \(\displaystyle \int \frac{4x^2-3x+1}{(2x+1)(x+2)(x-3)}dx\text{.}\)
