can u help me w/ the following questions that im not still clear about..the problem are did but can u explain?

Question

some points i mentioned? tnks

http://i181.photobucket.com/albums/x85/ctti_3/Untitled-86.jpg

Answer 1

You're in good shape until the evaluation phase

∫ 2 lnx dx....................x=1 to 4e

= 2 * [ xlnx - x ].................evaluated at 4e and 1 and subtract

= 2 * [ (4e)*ln(4e) - (4e) - (1*ln1 -1) ]

= 2 * [ (4e)*ln(4e) - (4e) + 1 ]........This is where the +1 appears because 1*ln1=0 and, watch those signs -(-1) = +1

= 2 * [ (4e)*[ ln(4) + ln(e) ] - 4e + 1 ]......since ln(4e)=ln(4)+ln(e)

= 2 * [ 4e*ln(4) + 4e - 4e + 1 ] ..........since ln(e)=1

= 2 * [ 4e*ln(4) + 1 ]

= 8e*ln(4) + 2

Answer 2

As far as i see ln(4e) is the ln of a product = ln4+lne and ln e=1
so ln(4e) =ln4 +1