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[unparseable or potentially dangerous latex formula] IS NOT EQUAL TO [unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]

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[unparseable or potentially dangerous latex formula]

SO,

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SO,

[unparseable or potentially dangerous latex formula]

@ SXK: PLEASE DON'T SAY THE ANSWER, FOR YOU SENT ME THIS...

The integration by parts equation is really just the integral of the product rule, which means it is only true up to a constant. Usually, the method is still useful and we can ignore the constant as long as the two integrals are of different functions. However, there technically should be a "C" in the integration by parts equation, so we really have 0=-1+C, which is perfectly ok. In other words, the two integrals of tangent don't have to be equal, they can differ by a constant.

Thanks! These problems are really fun to think through.

I love these. Next one.
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
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Is it b/c [unparseable or potentially dangerous latex formula]

only applies when a & b are both positive numbers?

If thats not right, then I have no clue. :P

[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]

You can't do that because the reciprocal of i is -i.

I guess this is an easy one to figure out... @nsguy,sxk -> DON'T TELL THE ANS. CUZ UVE ALREADY SEEN IT... sxk sent this to me earlier...

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[unparseable or potentially dangerous latex formula]

You can't divide by zero, hence the fallacy is within the third equality, since [unparseable or potentially dangerous latex formula] is zero.