∫ 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
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