Quick question

What's the inverse chain rule?

My brain isn't working properly today, and while revising c4 I managed to confuse myself.

Why can't you integrate for example sin^2 (x) using inverse chain rule, ie increase power by 1, divide by new power, divide by differential of sinx to get (tanx sin^2 x)/3 ?

I think that's what they call integration by substitution

I think he might mean by "by inspection." But don't hold me to it.

You can't start dividing by functions of x, you can only adjust numerical factors. You can also check if you're wrong. Try differentiating what you think you'd get. You don't get back to what you want.

Hmm so you can't divide by functions of x when you integrate but you can multiply by functions of x when you differentiate? weird; P

What?

What I mean is you can only adjust numerical factors. When you integrate, sometimes you can do it by inspection and use a little bit of prior knowledge to "guess" the derivative. You then check by differentiating and then adjust the numerical factor. For example, if you differentiate and end up with a 2 in front, but you need a 3, you adjust the numerical factor to (3/2) to give you this. But if you end up with an extra "x" term, which you don't have in your integral, you can't put a (1/x) in front of it.