I believe your idea isIn my easy calculus puzzle, I've submitted (in another form) and I pointed out how by letting 1 = would lead to a solution. The formula can be rewritten as 1 = for the numerator of the first fraction which of course leads to the same answer.
That method I refer to as the addition-subtraction method or ASM for short (which is applied to the numerators of fractions). This is a method that should be taught in calculus classes and demonstrated in texts, but unfortunately isn't. It's a powerful method for simplifying fractions and I'll now give you another example in the form of an easy puzzle.
Using ASM, transform into two fractions and then go on to integrate those two fractions.