
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!

Re: Integral by parts
All he did was factor out a $\displaystyle 2y^{\frac{1}{2}}$ from the first two terms and then recall $\displaystyle y^{\frac{1}{2}} = \sqrt{y}$. It was for simplification only

Re: Integral by parts
Let's see, I'm just gonna go from the first bold line to the line above it:
$\displaystyle 2y^{1/2}\cdot (\ln(y)2)+C=\ln(y)\cdot 2y^{1/2}2\cdot2y^{1/2} +C = 2y^{1/2}\cdot \ln(y)4y^{1/2} +C $
and the last bold line is just because by definition $\displaystyle y^{1/2}=\sqrt{y}$.

Re: Integral by parts
OH! I see now. All he did was factor...wow. I appreciate both of your help!