How to Write the Integral in Different Forms

Question

I need to write the integral $\int dx\sqrt{-x^2 + 6x -5}$ in the form $\int dx\sqrt{b^2-(x-a)^2}$

I'm assuming that -1 is the a value and 6 is the b value?

Answer 1

First: $-x^2 + 6x - 5 = -(x^2 - 6x + 5)$

Complete the square: $= -[(x^2 - 6x + 9) - 9 + 5] = -[(x - 3)^2 - 4]$

Then multiply $-1$ into the bracket: $= 4 - (x - 3)^2 = 2^2 - (x - 3)^2$