![]() We have just swapped x and theta, but I have solved the problem so that you can see that it will be the same. Scenario TWO: This is identical to how Sal solves the problem in the video. Regardless, the curves for -arccos(x / sqrt(u)) and arcsin(x / sqrt(u)) are identical except that one is vertical translation of the other, and that is all we are trying to prove with integration, anyway. ![]() Since after we integrate we are left with a constant of integration (I have used c to denote said constant), this "absorbs" the extra pi / 2 that we would need to add for the graphs to match exactly. Although this appears different, if you look at this graph (drag the slider at the top-left at your own leisure) => <= then you will see that -arccos(x / sqrt(u)) + pi/2 is actually equal to arcsin(x / sqrt(u)). If we choose x as the side adjacent to theta, then we will end up with -arccos(x / sqrt(u)) + c. Scenario ONE: this one is comparatively complex, but it still does make sense. ![]() There are two scenarios, and these are as follows: ![]()
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