Section 5.6 Partial fractions (TI6)
Learning Outcomes
I can integrate functions using the method of partial fractions.
Activity 5.6.1.
Using the method of substitution, which of these is the integral of \(\displaystyle\int \frac{1}{(ax+b)} dx\text{?}\)
\(\displaystyle \frac{1}{a}\ln(ax+b) +C\)
\(\displaystyle \frac{1}{a}\ln|ax+b| +C\)
\(\displaystyle \frac{1}{x}(\ln(ax)+\ln(b))+C\)
\(\displaystyle \frac{1}{a}(\ln|ax|+\ln|b|)+C\)
Activity 5.6.2.
Which of the following is equivalent to the rational expression \(\displaystyle\frac{3x+3}{2x^2+3x}\text{?}\)
\(\displaystyle \frac{3}{2x}+\frac{1}{x}\)
\(\displaystyle \frac{3}{x}+\frac{3}{2x+3}\)
\(\displaystyle \frac{1}{x}+\frac{1}{2x+3}\)
\(\displaystyle \frac{3}{2x^2}+1\)
Activity 5.6.3.
Based on your choice in Activity 5.6.2, which of these is the integral for \(\displaystyle\int \frac{3x+3}{2x^2+3x} dx\text{?}\)
\(\displaystyle \frac{3}{2}\ln|x|+\ln|x|+C\)
\(\displaystyle \ln|x|+\frac{1}{2}\ln|2x+3|+C\)
\(\displaystyle -\frac{3}{2}x^{-1}+x+C\)
\(\displaystyle 3\ln|x|+\frac{3}{2}\ln|2x+3|+C\)
Activity 5.6.4.
Which of the following is an antiderivative for \(\displaystyle\frac{1}{(x+k)^2+h^2}\text{?}\)
\(\displaystyle \arctan(x)\)
\(\displaystyle \arctan(x+k)\)
\(\displaystyle \frac{1}{h}\arctan\left(\frac{x+h}{h}\right)\)
\(\displaystyle \frac{1}{h^2}\arctan\left(\frac{x+k}{h^2}\right)\)
\(\displaystyle \frac{1}{h}\arctan\left(\frac{x+k}{h}\right)\)
Activity 5.6.5.
Consider the irreducible quadratic \(x^2-4x+80\text{.}\) Which of the following is the completed square form of this quadratic expression?
\(\displaystyle (x-4)^2+80\)
\(\displaystyle (x-2)^2+80\)
\(\displaystyle (x-4)^2+64\)
\(\displaystyle (x-2)^2+76\)
\(\displaystyle (x-4)^2+76\)
Activity 5.6.6.
Using the fact that \(\displaystyle\int \frac{1}{x^2-8x+80} dx = \int \frac{1}{(x-4)^2+64} dx\text{,}\) evaluate the integral \(\displaystyle \int \frac{1}{x^2-8x+80} dx\text{.}\)
Activity 5.6.7.
Suppose that one could write \(\displaystyle\frac{4x^2-5x-2}{x^2(x-2)} =\frac{A}{x}+\frac{B}{x^2}+\frac{C}{x-2}\text{.}\)
Recall that:
What are the values of \(A, B\) and \(C\text{?}\)
\(\displaystyle A=4, B=-5, C=-2\)
\(\displaystyle A=3, B=1, C=1\)
\(\displaystyle A=1, B=1, C=1\)
\(\displaystyle A=1, B=3, C=-1\)
\(\displaystyle A=2, B=1, C=1\)
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 \(q\text{.}\)
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Linear Terms: Let \((x-a)\) divide \(q(x)\text{,}\) where \((x-a)^n\) is the highest power of \((x-a)\) that divides \(q(x)\text{.}\) Then the decomposition of \(\frac{p(x)}{q(x)}\) will contain the sum
\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 \(x^2+bx+c\) be an irreducible quadratic that divides \(q(x)\text{,}\) where \((x^2+bx+c)^n\) is the highest power of \(x^2+bx+c\) that divides \(q(x)\text{.}\) Then the decomposition of \(\frac{p(x)}{q(x)}\) will contain the sum
\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*}
Activity 5.6.9.
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 5.6.10.
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 5.6.11.
Consider that the partial decomposition of \(\displaystyle \frac{x^2+5x+3}{(x+1)^2x}\) is
What equality do we obtain if we multiply both sides of the above equation by \((x+1)^2x\text{?}\)
\(\displaystyle x^2+5x+3=Ax(x+1)+Bx+C(x+1)^2\)
\(\displaystyle x^2+5x+3=A(x+1)+B(x+1)^2+Cx\)
\(\displaystyle x^2+5x+3=Ax(x+1)+Bx+C(x+1)\)
\(\displaystyle x^2+5x+3=A(x+1)+Bx+Cx^2\)
\(\displaystyle x^2+5x+3=Ax(x+1)+Bx^2+C(x+1)^2\)
Activity 5.6.12.
Notice that \(x^2+5x+3=Ax(x+1)+Bx+C(x+1)^2\text{.}\) Which of the following values can we determine by setting \(x=0\text{?}\) Select all that apply. \(\displaystyle A\) \(\displaystyle B\) \(\displaystyle C\) Which of the following values can we determine by setting \(x=-1\text{?}\) Select all that apply. \(\displaystyle A\) \(\displaystyle B\) \(\displaystyle C\)(a)
(b)
Activity 5.6.13.
Rewrite \(Ax(x+1)+x+3(x+1)^2\) so that \(Ax(x+1)+x+3(x+1)^2=?x^2+?x+?\text{.}\) Knowing that
can we solve for \(A\text{?}\)
Activity 5.6.14.
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 5.6.12 and Activity 5.6.13, evaluate \(\displaystyle \int \frac{x^2+5x+3}{(x+1)^2x} dx\text{.}\)
Activity 5.6.15.
Given that \(\displaystyle\frac{x^3-7x^2-7x+15}{x^3(x+5)}=\frac{A}{x}+\frac{B}{x^2}+ \frac{C}{x^3}+\frac{D}{x+5}\text{,}\) what are \(A, B, C\text{,}\) and \(D\text{?}\)
\(\displaystyle A=1, B=-7 C=-7, D=15\)
\(\displaystyle A=-1, B=-2, C=3, D=2\)
\(\displaystyle A=1, B=-2, C=-3, D=2\)
\(\displaystyle A=-1, B=2, C=3, D=-2\)
Activity 5.6.16.
Given your choice in Activity 5.6.15 Find \(\displaystyle\int \frac{x^3-7x^2-7x+15}{x^3(x+5)} dx.\)
Activity 5.6.17.
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 5.6.18.
Given your choice in Activity 5.6.17 Find \(\displaystyle\int \frac{2x^3+2x+4}{x^4+2x^3+4x^2} dx\text{.}\)
Activity 5.6.19.
Given that \(\displaystyle \frac{2x+5}{x^2+3x+2}=\frac{-1}{x+2}+\frac{3}{x+1}\text{,}\) find \(\displaystyle\int_0^3 \frac{2x+5}{x^2+3x+2} dx\text{.}\)
Remark 5.6.20.
This is all well and good, but given a rational function, how might one actually find coefficients for the partial fractions if we weren't given them?
Activity 5.6.21.
Evaluate \(\displaystyle \int \frac{4x^2-3x+1}{(2x+1)(x+2)(x-3)}dx\text{.}\)