Use integration by parts to integrate Integration sign, x^5 ln (x) dx

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jeew-m | College Teacher | (Level 1) Educator Emeritus

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Using integration by parts;

intudv = uv-intvdu

Let v=x^6 and u=lnx


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`

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beckden | High School Teacher | (Level 1) Educator

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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.

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