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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?
  1. \(\displaystyle u=x^3\)
  2. \(\displaystyle u=x^3+x\)
  3. \(\displaystyle u=x^2+x+1\)
  4. Substitution is not effective

Activity 5.6.2.

Using the method of substitution, which of these is equal to \(\displaystyle\int \frac{5}{x+7} dx\text{?}\)
  1. \(\displaystyle 5\ln|x+7| +C\)
  2. \(\displaystyle \frac{5}{7}\ln|x+7| +C\)
  3. \(\displaystyle 5\ln|x|+5\ln|7|+C\)
  4. \(\displaystyle \frac{5}{7}\ln|x|+C\)

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 \(\displaystyle\frac{1}{x}+\frac{1}{x^2+1}\text{?}\)
  1. \(\displaystyle \frac{2x}{x^2+x+1}\)
  2. \(\displaystyle \frac{x^3+x}{x^2+x+1}\)
  3. \(\displaystyle \frac{2x}{x^3+x}\)
  4. \(\displaystyle \frac{x^2+x+1}{x^3+x}\)

Activity 5.6.5.

Based on the previous activities, which of these is equal to \(\displaystyle\int \frac{x^2+x+1}{x^3+x} dx\text{?}\)
  1. \(\displaystyle \ln|x|+\arctan(x)+C\)
  2. \(\displaystyle \ln|x^2+x+1|+C\)
  3. \(\displaystyle \ln|x^3+x|+C\)
  4. \(\displaystyle \arctan(x^3+x)+C\)

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 \(\int\frac{10x-11}{x^2+x-2}\text{?}\)
  1. \(\displaystyle 7\ln|x-1|+3\arctan(x+2)+C\)
  2. \(\displaystyle 7\ln|x-1|+3\ln|x+2|+C\)
  3. \(\displaystyle 7\arctan(x-1)+3\arctan(x+2)+C\)
  4. \(\displaystyle 7\arctan(x-1)+3\ln|x+2|+C\)

Observation 5.6.7.

To find integrals like \(\int \frac{x^2+x+1}{x^3+x} dx\) and \(\int\frac{10x-11}{x^2+x-2}\text{,}\) 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*}

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 \(\displaystyle\frac{x^3-7x^2-7x+15}{x^3(x+5)}\text{?}\)
  1. \(\displaystyle \frac{A}{x}+\frac{B}{x+5}\)
  2. \(\displaystyle \frac{A}{x^3}+\frac{B}{x+5}\)
  3. \(\displaystyle \frac{A}{x}+\frac{B}{x^2}+ \frac{C}{x^3}+\frac{D}{x+5}\)
  4. \(\displaystyle \frac{A}{x}+\frac{B}{x^2}+ \frac{C}{x^3}+\frac{Dx+E}{x+5}\)

Activity 5.6.11.

Which of the following is the form of the partial fraction decomposition of \(\displaystyle\frac{x^2+1}{(x-3)^2(x^2+4)^2}\text{?}\)
  1. \(\displaystyle \frac{A}{x-3}+\frac{B}{(x-3)^2}+\frac{C}{x^2+4}+\frac{D}{(x^2+4)^2}\)
  2. \(\displaystyle \frac{A}{x-3}+\frac{B}{(x-3)^2}+\frac{Cx+D}{(x^2+4)^2}\)
  3. \(\displaystyle \frac{A}{x-3}+\frac{B}{(x-3)^2}+\frac{C}{x^2+4}+\frac{Dx+E}{(x^2+4)^2}\)
  4. \(\displaystyle \frac{A}{x-3}+\frac{B}{(x-3)^2}+\frac{Cx+D}{x^2+4}+\frac{Ex+F}{(x^2+4)^2}\)

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 \((x+1)^2x\text{?}\)
  1. \(\displaystyle x^2+5x+3=Ax(x+1)+Bx+C(x+1)^2\)
  2. \(\displaystyle x^2+5x+3=A(x+1)+B(x+1)^2+Cx\)
  3. \(\displaystyle x^2+5x+3=Ax(x+1)+Bx+C(x+1)\)
  4. \(\displaystyle x^2+5x+3=Ax(x+1)+Bx^2+C(x+1)^2\)

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:
  1. \(\displaystyle A=1\)
  2. \(\displaystyle B=-2\)
  3. \(\displaystyle C=3\)
(b)
By substituting \(x=-1\) into the equation, we may find:
  1. \(\displaystyle A=-4\)
  2. \(\displaystyle B=1\)
  3. \(\displaystyle C=5\)

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?
  1. \(\displaystyle -2\)
  2. \(\displaystyle 3\)
  3. \(\displaystyle 4\)
  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?
  1. \(\displaystyle \frac{1}{x}+\frac{1}{x^2}+\frac{2x-1}{x^2+2x+4}\)
  2. \(\displaystyle \frac{2}{x}+\frac{0}{x^2}+\frac{-1}{x^2+2x+4}\)
  3. \(\displaystyle \frac{0}{x}+\frac{1}{x^2}+\frac{-1}{x^2+2x+4}\)
  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 \(\displaystyle \int \frac{4x^2-3x+1}{(2x+1)(x+2)(x-3)}dx\text{.}\)

Subsection 5.6.2 Videos

Figure 110. Video: I can integrate functions using the method of partial fractions