Find appropriate substitution for indefinite integral.

Question

Find the indefinite integral,

$$ I = \int \frac{8 - 2x}{\sqrt{6x - x^2}} dx\,. $$

I know this is a 'substitution' question, but I can't work out what to substitute. Please could you tell me the substitution and how you got to it.

Answer 1

use the substitution $3 \sin t = x - 3$ which comes from completing the square of $6x - x^2$ and looking ahead so that you can get rid of the square root in the denominator.