10.9#3

This problem won't work. the integral of -25sin(5t) is 5cos(5t). But it keeps saying that is the incorrect answer, and therefore I can't get the antiderivative of that either.

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Well, you say that "the integral of -25sin(5t) is 5cos(5t)" , but this isn't true.  AN integral is 5cos(5t), but so is 5cos(5t)+K where K is any constant.  You've reflexively chosen K=0, but any other constant will also be an integral of -25sin(5t).  But this value of K gives the wrong value for v(0), as you can check for yourself.  Which is the right value?