Okay, so I was on the correct path. I ended up with ,
but wasn't sure if it was it was correct; now, obviously, it is correct, but I can't seem to figure out why it is correct. I am just trying to justify every step, but it just seems unusual that the x's cancel out and stuff like that.
I figured by doing the u-substitution I would have one sure-fire way of doing, and that I would also see how the derivative and integrals relate in the process; so, now, I am just trying to make sense of the u-substitution in each integration case.
In this case, you have
Can you see that if we use as the inner function, its derivative is ? So the integrand is of the form we require.
So we'd let , and note that when and when , then the integral becomes
which is very easy to solve
Another way to go about it:
Integrating indefinitely yields:
And because the Cs cancel anyway:
This is how I've been taught to approach an example like this, I only change the upper and lower limits if substituting x back into the expression would make it unmanageable messy. Also, it's not so uncommon for all the Xs to cancel out of an example - when I learned how to integrate by substitution all the examples simplified greatly with the correct substitution.
And, just in case a picture helps...
... where (key in spoiler) ...
Don't integrate - balloontegrate!
Balloon Calculus; standard integrals, derivatives and methods
Balloon Calculus Drawing with LaTeX and Asymptote!