Find the antiderivative of x^1/2/(x-1)

F(x) is such a function such as dF/dx =
f(x).

We'll calculate the function F(x) integrating the
given function:

Int f(x)dx = F(x) +
C

Int sqrtx dx/(x-1)

We'll
write the denominator of the function as a difference of
squares:

x - 1 = (sqrtx - 1)(sqrtx +
1)

We'll re-write the
integral:

Int sqrtx dx/(sqrtx - 1)(sqrtx +
1)

We'll add and subtract 1 to the
numerator:

Int (sqrtx + 1 - 1 )dx/(sqrtx - 1)(sqrtx +
1)

We'll re-group the terms in a convenient
way:

Int (sqrtx + 1 - 1 )dx/(sqrtx - 1)(sqrtx + 1) =
Int(sqrtx + 1)dx/(sqrtx - 1)(sqrtx + 1) - Int dx/(sqrtx - 1)(sqrtx +
1)

We'll simplify and we'll
get:

Int (sqrtx + 1 - 1 )dx/(sqrtx - 1)(sqrtx + 1) = Int dx
- Int dx/(x-1)

Int f(x)dx = x - ln|x-1| +
C

F(x) = x - ln|x-1| +
C