First, here's the formula for "integration by parts":
int f'(x) * g(x) "d"x = f(x) g(x) - int f(x) g'(x) "d" x
Now, in your product x * sqrt(x-5) you need to determine which factor is f'(x) and which one is g(x) since you will need to integrate one of them and differentiate the other one.
As it doesn't make a big difference with the radical expression but would be a good thing to differentiate x, let's assume that f(x) = x and g'(x) = sqrt(x-5).
As next, you need to differentiate f(x) and integrate g'(x):
f(x) = x => f'(x) = 1
g'(x) = sqrt(x-5) = (x-5) ^(1/2) => g(x) = 2/3 * (x-5)^(3/2)
Now we can use the formula:
int x * sqrt(x-5) "d"x
= x * 2/3 * (x-5)^(3/2) - int 1 * 2/3 * (x-5)^(3/2) "d"x
= 2/3 x (x-5)^(3/2) - 2/3 int (x-5)^(3/2) "d"x
= 2/3 x (x-5)^(3/2) - 2/3 *2/5 (x-5)^(5/2)
= 2/3 x (x-5)^(3/2) - 4/15 (x-5)^(5/2)
Hope that this helped!
