`int (sqrt(x)/(sqrt(x) - 3)) dx` Find the indefinite integral by u substitution. (let u be the denominator of the integral)

Expert Answers

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

To apply u-substitution , we let `u = sqrt(x)-3` .

Then  ` du = 1/(2sqrt(x) dx` .

Rearrange  `du = 1/(2sqrt(x)) dx` into `dx =2sqrt(x) du`

Substituting `dx=2sqrt(x) du` and `u =sqrt(x)-3` :

`int sqrt(x)/(sqrt(x)-3)dx = int sqrt(x)/u*2sqrt(x) dx`        

Simplify: `sqrt(x)*sqrt(x) = x`

`int sqrt(x)/u *2sqrt(x) du = int (2x)/u du`

Rearrange `u=sqrt(x)-3` into `sqrt(x)=u+3`

Squaring both sides of`sqrt(x)=u+3` then

`x=u^2+6u+9`

`int (2x)/u du = 2 int (u^2+6u+9)/u du`

                   `= 2 int (u^2/u + 6u/u + 9/u) du`

                   `= 2 int (u + 6 + 9/u) du `

                   `=2 *(u^2/2+6u+9lnabs|u|) +C`

  Substitute u =sqrt(x)-3:

`2 *(u^2/2+6u+9ln|u|)+C =2 *((sqrt(x)-3)^2/2+6(sqrt(x)-3)+9ln|(sqrt(x)-3)|)+C`

                                   ` =(sqrt(x)-3)^2+12(sqrt(x)-3)+18ln|(sqrt(x)-3)| +C`

                                   ` = x-6sqrt(x)+9+12sqrt(x)-36 +18ln|sqrt(x)-3|+C`

                                  `= x + 6sqrt(x)-27 +18ln|sqrt(x)-3|+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