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:

**∫**sin(x)cos(x)dx

How would you go about doing it? Well, you can define U=sin(x), and rewrite it like this:

**∫**Ucos(x)dx

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:

**∫**Ucos(x)^{dU}⁄_{cos(x)}

and we find that, miraculously, the cos(x) values cancel out!

**∫**UdU = ½U^{2} = ½sin^{2}(x)

And we have our final answer. Why does this skulduggery work? who knows! I don’t!

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