Thread: Integral by parts

Integral from 9 to 16 lny/sqrt(y)

Here's my work and on the parts bolded I asked for help from someone else but Im not sure WHY he did those steps:

∫ ln(y)⋅y^(−½) dy =
2⋅y^(½)⋅ln(y) − ∫ 2⋅y^(½)⋅(1/y) dy
2⋅y^(½)⋅ln(y) − 2⋅∫ y^(−½) dy
2⋅y^(½)⋅ln(y) − 2⋅(2)⋅y^(½) + C
2⋅y^(½)⋅ln(y) − 4⋅y^(½) + C
2⋅y^(½)⋅[ ln(y) − 2 ] + C
2⋅√(y)⋅[ ln(y) − 2 ] + C

Im not sure why and how the 4y^1/2 got eliminated/broken up from the problem. Can anyone explain why these two parts happen? I can handle the rest of the problem from here as you just have to plug in the two intervals and subtract them from each other.
Thanks!

All he did was factor out a from the first two terms and then recall . It was for simplification only

Let's see, I'm just gonna go from the first bold line to the line above it:

and the last bold line is just because by definition .

OH! I see now. All he did was factor...wow. I appreciate both of your help!