`int p^5 ln p dp` Evaluate the integral

`int p^5 ln(p) dp`

To evaluate, apply integration by parts intu dv = uv  -int vdu.

So let

`u= ln (p)`

and

`dv = p^5 dp`

Then, differentiate u and integrate dv.

`du=1/p dp`

and

`v = int p^5dp = p^6/6`

And, plug-in them to the formula. So the integral becomes:

`int p^5 ln(p) dp`

`= ln(p) *p^6/6 - int p^6/6*1/pdp`

`= (p^6ln(p))/6 - int p^5/6 dp`

`= (p^6 ln(p))/6 - 1/6 int p^5 dp`

...

`int p^5 ln(p) dp`

To evaluate, apply integration by parts intu dv = uv  -int vdu.

So let

`u= ln (p)`

and

`dv = p^5 dp`

Then, differentiate u and integrate dv.

`du=1/p dp`

and

`v = int p^5dp = p^6/6`

And, plug-in them to the formula. So the integral becomes:

`int p^5 ln(p) dp`

`= ln(p) *p^6/6 - int p^6/6*1/pdp`

`= (p^6ln(p))/6 - int p^5/6 dp`

`= (p^6 ln(p))/6 - 1/6 int p^5 dp`

`=(p^6 ln(p))/6 - 1/6*p^6/6 + C`

`=(p^6ln(p))/6 -p^6/36+C`

Therefore,  `int p^5 ln(p) dp = (p^6 ln(p))/6 - p^6/36 + C` .