Integrals are such hacks

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 dydx 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)
dUdx = cos(x)

And we now simply rearrange the fraction to solve for dx:
dx = dUcos(x)

which we then substitute into our integral:

Ucos(x)dUcos(x)

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!

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