How do you integrate #(9x^2 + 1) /( x^2(x − 2)^2)# using partial fractions?

Question

Christopher Agresta · Accepted Answer

The answer is #=-1/(4x)-37/(4(x-2))+1/4ln(|x|)-1/4ln(|x-2|)+C#

We perform the decomposition into partial fractions

#(9x^2+1)/(x^2(x-2)^2)=A/(x^2)+B/(x)+C/(x-2)^2+D/(x-2)#

#=(A(x-2)^2+B(x(x-2)^2)+C(x^2)+D(x^2(x-2)))/(x^2(x-2)^2)#

As the denominators are the same, we compare the numerators

#9x^2+1=A(x-2)^2+B(x(x-2)^2)+C(x^2)+D(x^2(x-2))#

Let #x=0#, #=>#, #1=4A#, #=>#, #A=1/4#

Let #x=2#, #=>#, #37=4C#, #=>#, #C=37/4#

Coefficients of #x^2#

#9=A-4B+C-2D#

Coeficients of #x#,

#0=-4A+4B#, #=>#, #B=A=1/4#

#1/4-1+37/4-2D=9#

#2D=37/4-3/4-9=34/4-9=-2/4=-1/2#

#D=-1/4#

So,

#(9x^2+1)/(x^2(x-2)^2)=(1/4)/(x^2)+(1/4)/(x)+(37/4)/(x-2)^2+(-1/4)/(x-2)#

Therefore,

#int((9x^2+1)dx)/(x^2(x-2)^2)=1/4intdx/(x^2)+1/4intdx/(x)+37/4intdx/(x-2)^2-1/4intdx/(x-2)#

#=-1/(4x)-37/(4(x-2))+1/4ln(|x|)-1/4ln(|x-2|)+C#

Emma Aldridge · Answer

undefined To integrate $ \frac{9x^2 + 1}{x^2(x - 2)^2} $ using partial fractions, follow these steps:

1. Express the given fraction as the sum of simpler fractions using partial fraction decomposition.
2. Determine the constants for each term in the decomposition.
3. Integrate each partial fraction term separately.

Here are the steps:

1. Express $ \frac{9x^2 + 1}{x^2(x - 2)^2} $ as $ \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x - 2} + \frac{D}{(x - 2)^2} $.

2. Multiply both sides by the denominator $ x^2(x - 2)^2 $ to clear the fractions and solve for the constants $ A $, $ B $, $ C $, and $ D $.

3. After solving for $ A $, $ B $, $ C $, and $ D $, integrate each term separately.

The integral of $ \frac{A}{x} $ is $ A \ln|x| $.
   
   The integral of $ \frac{B}{x^2} $ is $ -\frac{B}{x} $.
   
   The integral of $ \frac{C}{x - 2} $ is $ C \ln|x - 2| $.
   
   The integral of $ \frac{D}{(x - 2)^2} $ is $ -\frac{D}{x - 2} $.

4. Combine the results from step 3 to obtain the final integral expression.

The integration process involves solving for the constants $ A $, $ B $, $ C $, and $ D $, then integrating each partial fraction term separately.