In my book, under the topic "Evaluation of Integrals by using Trigonometric Substitutions", it is given, in order to simplify integrals containing the expressions $\sqrt{\frac{x-\alpha}{\beta-x}}$ and $\sqrt{(x-\alpha)(\beta-x)}$, the substitution $x=\alpha\cos^2\theta+\beta\sin^2\theta$ must be used. If I remember this form and the substitution, then it definitely helps simplify the integrand. The first expression gets simplified to $\tan\theta$ and the second to $\sin\theta\cos\theta(\alpha-\beta)$.
I understand that if we do this kind of substitution we greatly simplify the expression. But how do we determine what to substitute in the first place or in other words, if I forget the substitution, is there any way to determine which substitution works well to simplify the integrand? How did the author figure out this substitution is the best fit for this kind of expression? Was it a guess or is there any mathematical reasoning behind it?
Best Answer
In the first place, one notices that the expression can be "normalized" by means of a linear transform that maps $\alpha,\beta$ to $-1,1$, giving the expressions
$$\sqrt{1-x^2}\text{ and }\sqrt{\frac{1+x}{1-x}}=\frac{1+x}{\sqrt{1-x^2}}.$$
Then the substitution $x=\cos\theta$ comes naturally. We could stop here.
Coming back to the unscaled originals, we have
$$x=\frac{\alpha+\beta+(\beta-\alpha)\cos\theta}2$$
which is also
$$x=\frac{(\alpha+\beta)(\cos^2\frac\theta2+\sin^2\frac\theta2)+(\beta-\alpha)(\cos^2\frac\theta2-\sin^2\frac\theta2)}2=\alpha\sin^2\frac\theta2+\beta\cos^2\frac\theta2.$$