Integral similar to Beta function

Question

I am trying to calculate the integral $$\int_{1}^{2} y^{k-1}(y-1)^{-1/2}dy,$$ where $k$ is positive integers. But the integral domain is a little different from the definition of the Beta function. I have tried to replace $y$ by $y+1$, so we get following integral $$\int_{0}^{1} (y+1)^{k-1}y^{-1/2}dy.$$ It seems no help. I hope the integral above can be expressed in terms of the Beta function. Is it possible? Or there is a simplified result?

Answer 1

If you are familiar with the gaussian hypergeometric function, the definite integral is $$2\, \, _2F_1\left(\frac{1}{2},1-k;\frac{3}{2};-1\right)$$ This is identical to the result given by @Eric Towers.

Both formulae are valid for any value of $k$ (positive, negative, rational or not and complex)