Bit of give and take in the numerator...
Do parts on
(integrating the fraction)...
Edit: Just in case a picture helps...
... where (key in spoiler) ...
Don't integrate - balloontegrate!
Balloon Calculus; standard integrals, derivatives and methods
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What a strange integral... considering the amount of trouble you have to go through to solve it (using any "normal" method) you'd think this can't have a nice solution but it has (I used Mathematica - sorry if that feels for you like I cheated) and wound up with something that was... well, funny how "nice" it was.
Anyway, all you have to do is note that the form tom@ballooncalculus wrote can be rewritten as:
And that is practically the definition of the quotient rule, so:
Of course you get the same result by substituting
Now, since any other way I can think of would require evaluating
, I guess this is the way it's supposed to be solved. I also congratulate anyone who would have been able to do that .
one might guess, because the numerator is a square, that the integral may be some quotient.
, which we want to be equal to
that is, we want .
if f(x) is a polynomial, with leading term of degree n:
, then the leading term of
this tells us n = 1, and a = -1, that is f(x) = -x + b, for some real number b.
, which only holds when b = -1.
we can thus conclude that f(x) = -x - 1