`int (x^2+12x+12)/(x^3-4x) dx` Use partial fractions to find the indefinite integral

Expert Answers

An illustration of the letter 'A' in a speech bubbles


`int (x^2+12x+12)/(x^3-4x)dx `

To solve using partial fraction method, the denominator of the integrand should be factored.

`(x^2+12x+12)/(x^3-4x) =(x^2+12x+12)/(x(x-2)(x+2))`

Then, express it as sum of fractions.

`(x^2+12x+12)/(x(x-2)(x+2)) = A/x + B/(x-2) + C/(x+2)`

To determine the values of A, B and C, multiply both sides by the LCD of the fractions present.

`x(x-2)(x+2)*(x^2+12x+12)/(x(x-2)(x+2)) = (A/x + B/(x-2) + C/(x+2))*x(x-2)(x+2)`

`x^2+12x+12=A(x-2)(x+2) +Bx(x+2)+Cx(x-2)`

Then, assign values to x in which either x, x-2 or x+2 will become zero.

So, plug-in x=0 to get the value of A.





Also, plug-in x=2 to get the value of B.





And subsitute x=-2 to get the value of C.

`(-2)^2 + 12(-2)+12=A(-2-2)(-2+2)+B(-2)(-2+2)+C(-2)(-2-2)`




So the partial fraction decomposition of the integral is

`int (x^2+12x+12)/(x^3-4x)dx`

`= int (x^2+12x+12)/(x(x-2)(x+2))dx`

`= int(-3/x +5/(x-2)-1/(x+2))dx`

Then, express it as three integrals.

`= int-3/xdx + int 5/(x-2)dx - int 1/(x+2)dx`

`= -3int 1/xdx + 5int 1/(x-2)dx - int 1/(x+2)dx`

To take the integral, apply the formula `int 1/u du =ln|u|+C` .

`=-3ln|x| + 5ln|x-2|-ln|x+2|+C`


Therefore,`int (x^2+12x+12)/(x^3-4x)dx=-3ln|x| + 5ln|x-2|-ln|x+2|+C`.

Approved by eNotes Editorial Team
Soaring plane image

We’ll help your grades soar

Start your 48-hour free trial and unlock all the summaries, Q&A, and analyses you need to get better grades now.

  • 30,000+ book summaries
  • 20% study tools discount
  • Ad-free content
  • PDF downloads
  • 300,000+ answers
  • 5-star customer support
Start your 48-Hour Free Trial