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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