`int (x^3 - 4x^2 - 4x + 20)/(x^2 - 5) dx` Find the indefinite integral.

Expert Answers

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

`int (x^3-4x^2-4x+20)/(x^2-5)dx`

To solve, divide the numerator by the denominator (see attached figure).

`= int (x - 4 + x/(x^2-5))dx`

`= int xdx - int4dx + int x/(x^2-5)dx`

For the first integral, apply the formula `int x^ndx = x^(n+1)/(n+1)+` C.

For the second integral, apply the formula `int adx = ax + C` .

`= x^2/2 - 4x +C + int x/(x^2-5)dx`

For the third integral, apply u-substitution method.

Let

`u = x^2-5`

Differentiate u.

`du=2x dx`

`(du)/2 =xdx`

Plug-in them to the third integral.

`=x^2/2 - 4x + C + int 1/(x^2-5)*xdx`

`=x^2/2 - 4x + C + int 1/u *(du)/2`

`= x^2/2 - 4x + C + 1/2int 1/u du`

Then, apply the formula `int 1/xdx = ln|x| + ` C.

`=x^2/2-4x + 1/2ln|u| + C`

And, substitute back  `u = x^2-5` .

`=x^2/2 - 4x +1/2ln|x^2-5|+C`

 

Therefore,  `int (x^3-4x^2-4x + 20)/(x^2-5)dx = x^2/2 - 4x + 1/2ln|x^2-5|+C` .

Images:
This image has been Flagged as inappropriate Click to unflag
Image (1 of 1)
Approved by eNotes Editorial Team

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