Allow me to explain my new theory, The "Mancini conjecture." Ok....lets say I have an integral like (4-x^2)^(1/2) dx. and letting u = 4-x^2, we get du/dx = -2x, and if I took the second derivative of du/dx....i would get -2 this would be ideal, because I would then have du'' = -2 dx, or -1/2 du'' = dx. So what I'm trying to ask is, if a u substitution doesnt work out clean...that is, when i take du/dx and I end up with something that doesnt help me, what exactly is stopping me from going on, that is keep taking the next derivative...just keep taking the nth derivative until eventually I end up with a constant, which I can then pull out in front of the integral. Of course, if I do this, I would have to compensate for this somehow. I am having difficulty seeing how I would compensate for this, but I have a feeling like there is an obvious compensation that could be made. Maybe take the antiderivative of the answer you get using du'' instead of du. Could I just add an another integral sign each time i differentiate u another time? that is take the antiderivative of an antiderivative as many times as i differentiate u? Maybe I would have to differentiate u again each time I take the next derivative of du/dx . Is the integral of u du the same as the integral of u' du''? I have this feeling like I'm on to something, because if the exponent is positive, wouldn't I eventually get a constant after n differentiations? Wouldn't this make taking integrals super easy? Just differentiate n times until you get a constant, apply n compensations to the integral for doing that, and pull the constant out front. So once I figure out how to relate the integral of u du'' to u du, I will claim my nobel prize. My intuition is "if you take the derivative of du n times, just take the nth antiderivative of your answer." I just thought of it today doing my calculus 2 homework, and I became excited. We haven't learned integration by parts yet. I should sit down with a pencil and paper and test my intuition, but i'm a B student and i suck at integrals, and USA mens hockey is on soon. Let me know if I'm on to something. Thanks!