Evaluate the following definite integral

Question

Evaluate the following definite integral:

 

∫(1 to 3) x²*ln*x dx

Answer 1

See the video below for a full solution:

 

http://www.youtube.com/watch?v=RHocN-VU5ak

 

9ln(3)-26/9=6.999

Answer 2

Do integration by parts:

u = lnx          v = x³/3

du = 1/x       dv = x² 

 

we have [ln(x)*(x³/3)] - ∫x³/3x   where both are defined from 1 to 3

 

Now plug in/simplify:

[ln(x)*(x3/3)] - ∫x3/3x

9ln(3) -∫x²/3

9ln(3) - [x³/9]    evaluation from 1 to 3

9ln(3) - 3 + 1/9

 

Which simplifies to 6.999