Using integration by parts;

intudv = uv-intvdu

Let v=x^6 and u=lnx

Then;

dv = 6x^5 dx

du = 1/x * dx

`intudv = int(lnx*6x^5)dx`

`int(lnx*6x^5)dx= (lnx)(x^6)-int(x^6)*1/xdx`

`int(6*lnx*x^5)dx = (lnx)(x^6)-intx^5dx`

`int(x^5*lnx) = (1/6)[(lnx)(x^6)-x^6/6]+C`

Use LIPET (ln, inv trig, polynomial, exponential, trig) to find u, we use `u=ln(x)` , so `dv=x^5.dx`

`du = 1/x dx` , `v = x^6/6 `

we use `int u dv = uv - int v du`

`int x^5 ln(x) dx = ln(x) x^6/6 - int x^6/6 1/x dx`

`int x^5 ln(x) dx = (x^6 ln(x))/6 - int x^5/6 dx`

`int x^5 ln(x) dx = (x^6 ln(x))/6 - 1/6 x^6/6 + C`

And our final answer is:

`int x^5 ln(x) dx = x^6/6(ln(x) - 1/6) + C`

You can check by taking the derivative (use the product property) of our answer.

## 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

Already a member? Log in here.

Are you a teacher? Sign up now