Find the following indefinite integrals, identifying any general rules of calculus that you see, 1) int (x^-2/3 ) ln(5x) dx

Expert Answers

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

You need to use integration by parts, hence you should remember the formula such that:

`int udv = uv - int vdu`

`dv= x^(-2/3) =gtv = x^(-2/3+ 1)/(-2/3+1)=gtv = 3x^(1/3)`

`u= ln 5x =gt du= (dx)/x`

`int x^(-2/3)*ln 5x dx =root(3)x*ln 5x - 3int x^(1/3)/x dx`

`int x^(-2/3)* ln 5x dx = root(3)x* ln 5x - 3int x^(1/3-1) dx`

`int x^(-2/3)* ln 5x dx = root(3)x* ln 5x - 3int x^(-2/3) dx`

`int x^(-2/3)* ln 5x dx = root(3)x* ln 5x - 9 root(3) x + c`

Hence, evaluating the integral using parts yields `int x^(-2/3)* ln 5x dx = root(3)x* ln 5x - 9 root(3) x + c.`

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