I’d like to take a moment to marvel/gripe about how cheaty some kinds of integration can be. If you know any calculus, you know that dy⁄dx isn’t supposed to be a fraction, and that treating it as such can lead you astray. Then we get taught this kind of thing.
Say, for example, you want to find this integral:
How would you go about doing it? Well, you can define U=sin(x), and rewrite it like this:
But now it’s no longer in terms of x, so we can’t integrate in terms of x. Well, no worries – we simply find dx in terms of dU. How?
U = sin(x)
dU⁄dx = cos(x)
And we now simply rearrange the fraction to solve for dx:
dx = dU⁄cos(x)
which we then substitute into our integral:
and we find that, miraculously, the cos(x) values cancel out!
∫UdU = ½U2 = ½sin2(x)
And we have our final answer. Why does this skulduggery work? who knows! I don’t!