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.
Plato, you should not discourage a student from using a correct method (especially one that is very straightforward, such as a u substitution), especially if it's the method the student has been told to use. Yes, it would be nice if a student can recognise what particular derivatives look like, and therefore what their antiderivatives look like, but that's an unrealistic expectation, as it requires students to remember far too much. It's much better to have students remember a single process than an uncountable number of derivatives.
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.
Well like I said, a good strategy is to look for an inner function and see if its derivative is a multiple.
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:
let
Integrating indefinitely yields:
So...
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) ...
Spoiler:
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Don't integrate - balloontegrate!
Balloon Calculus; standard integrals, derivatives and methods
Balloon Calculus Drawing with LaTeX and Asymptote!