Integration with respect to a function of x

You might think: what the hell are you talking about?

I’m talking about this:

Looks crazy, right? After this lesson it’s gonna look cool, trust me!

Let’s calculate the integral of x with respect to x².

One day I thought: “what is the integral of a function of x with respect to x²? That’s impossible I guess.” A couple of months later, that is less than a week ago, I realised: “substitution!”. If we know how to integrate with respect to x, u, t or whatever letter it is, then let’s let x² be equal to another letter, like α, (it can be any letter), and substitute the x inside the function with sqrt(α), since x²=α. Let me show you:

That’s it! Let’s make things more interesting:

Let’s perform integration by parts:

Let’s perform a substitution:

Let’s undo the substitutions:

Not that hard, is it?

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