# int_1^3 r^3 ln(r) dr Evaluate the integral

## Expert Answers int_1^3 r^3 ln(r) dr

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

So let

u = ln r

and

dv = r^3 dr

Then, differentiate u and integrate dv.

u=1/r dr

and

v= int r^3 dr=r^4/4

Plug-in them to the formula. So the integral becomes:

int r^3 ln(r) dr

= ln (r)* r^4/4 - int r^4/4 * 1/rdr

= (r^4 ln(r))/4 - 1/4 int r^3 dr

= (r^4 ln(r))/4 - 1/4*r^4/4

=(r^4 ln(r))/4 - r^4/16

And, substitute the limits of the integral.

int_1^3 r^3 ln(r) dr

= ((r^4ln(r))/4 - r^4/16) |_1^3

= ( (3^4ln(3))/4 - 3^4/16) - ((1^4ln(1))/4-1^4/16)

= (3^4 ln(3))/4-3^4/16 +1/16

= (81ln(3))/4-81/16+1/16

=(81ln(3))/4-80/16

=(81ln(3))/4-5

Therefore,  int_1^3 r^3 ln(r) dr = (81ln(3))/4-5 .

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