∫ x^3 *ln(2x) (from 2->1) can be written in the form pln 2 + q, where p and q are rational numbers. Find p and q.

Firstly I would think about how I could integrate this. In this case you would need integrate by parts. I use LATE to decide which value is u and v' which stands for Log,Algebra,Trig,Exp. Which ever one appears first in the order is made to be u and the other v'. In this case v=ln(2x) and u'=x³ . Next I would find v' and u. Giving v'=1/x and u=x⁴/4. Putting this into the formula gives:
x⁴/4*ln(2x)-∫1/x * x⁴/4 dx = (x⁴ln(2x))/4-∫x³/4 dxNext I would use the range the question asks for (being 2->1) which results in:[(x⁴ln(2x))/4]²₁-1/4[1/4 * x⁴]²₁Calculating this gives (16ln(4)/4)-(ln(2)/4) -1/16[(16-1)]=8ln(2)-1/4ln(2)-15/16=31/4 * ln(2)-15/16This is in the form the question asks for giving the answer p=31/4 and q=-15/16