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In general, we have root ax^2+bx+c accompanied by function of the form px+q..In those cases the general idea is to express px+q as a combination of the derivative of the quadratic and a constant. This enables you to break the given integral in two standard parts. This is the general idea behind this manipulation..
However when you have f(x, root ax^2+bx+c) as shown in the example, it is difficult to replace the x in terms of t if you have some general substitution like ax^2+bx+c=t etc.
I am attaching an image to show you how this first substitution helps you to reduce it down to some general form. Let me know if you have any problems in those steps. But if you understand them, try the other two substitutions to understand the essence of this theory.