AntiderivativeFind the antiderivative of x^1/2/(x-1).

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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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

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