I have to use the technique of integration by parts to evaluate the integrals. I'm having trouble with a particular problem:
$$ \int x (\sqrt{x+2}) dx $$
I'm using u-substitution, but since I'm also integrating by parts, my "u-substitution" will be using the variable "g"
$g=x+2$
$\frac {dg}{dx} = 1 dx = dx$
Now the original problem is:
$$ \int x (\sqrt{g}) dx = \int x g^\frac 12 dx $$
So, I begin integrating by parts:
$$ = (x)(\frac 23 g^\frac 32) -\int (\frac 23 g^\frac 32)(dx) $$
I take out the constant, then find the integral:
$$ = (x)(\frac 23 g^\frac 32) - (\frac 23) (\frac 25 g^\frac 52) + C $$
Which then gives me:
$$ = (\frac {2x}3 g^\frac 32) - (\frac 4{15} g^\frac 52) + C $$
This is where I'm stuck, and I feel that I've missed an early step in the substitution (something to do with x = u - 1?).
The answer I should eventually arrive at is:
$$ = \frac 2{15} (x+2)^\frac 32 (3x-4)+C $$