Integrating using inverse functions

Question

Someone the other day told me about the idea of evaluating integrals using horizontal instead of vertical bars (apparently something to do with Lebesgue integration but that's way too complicated for me to understand). So I was thinking about this and it occurred to me that an inverse function flips a function that in a way taking its integral is taking the original integral but horizontally. This then led me to come up with 2 identities I'm not sure are correct and I established graphically: $$ \int_0^xf(x)dx=xf(x)-\int_{f(0)}^{f(x)}f^{-1}(x)dx$$ or $$ \int f(x)dx=xf(x)-\int f^{-1}(f(x))df(x),$$ although I'm not sure if the second is valid notation but I think I've seen something like that written before somewhere. These are only valid in ranges where $f(x)$ is bijective. For example: $$\begin{align} \int \arcsin(x)dx &=x\arcsin(x)-\int \sin(\arcsin(x)d(\arcsin(x))\\ & = x\arcsin(x)+cos(\arcsin(x)) \\ & = x\arcsin(x)+\sqrt{1-x^2} \end{align}$$ is only valid for $-1 \le x\le 1$ which would be true anyway but would also be true if we evaluated the integral of $\sin(x)$ this way giving a range of $-\frac{\pi}{2}\le x\le \frac{\pi}{2}$ where the integral is valid. Having done a bit of reading around the subject of integrals I am sort of surprised why this is never taught as it makes some integrals incredibly easy even though they may already be possible. I just think it would be another tool in the arsenal of mathematicians, that I never see used. $$$$ tl;dr: is the identity valid and if so why is it never taught?

Answer 1

Here's a little proof. Hopefully this helps. $$I(x):=\int_0^x f^{-1}(a)\mathrm{d}a$$ Substitution $a=f(b)$: $$\therefore I(x)=\int_{f^{-1}(0)}^{f^{-1}(x)}bf'(b)\mathrm{d}b$$ Integration by parts: $$\mathrm{d}v=f'(b)\mathrm{d}b\Rightarrow v=f(b)\\u=b\Rightarrow \mathrm{d}u=\mathrm{d}b$$ $$\therefore I(x)=bf(b)\bigg|_{f^{-1}(0)}^{f^{-1}(x)}-\int_{f^{-1}(0)}^{f^{-1}(x)}f(b)\mathrm{d}b$$ $$\therefore I(x)=xf^{-1}(x)-\int_{f^{-1}(0)}^{f^{-1}(x)}f(b)\mathrm{d}b$$ $$\therefore \int_0^x f^{-1}(t)\mathrm{d}t=xf^{-1}(x)-\int_{f^{-1}(0)}^{f^{-1}(x)}f(t)\mathrm{d}t$$ QED

Example:

$$I=\int\arcsin x\ \mathrm{d}x$$ setting $f^{-1}(x)=\arcsin x$ gives $f(x)=\sin x$, and $F(x)=\int f(x)\mathrm{d}x=-\cos x$. We know that $$I=xf^{-1}(x)-(F\circ f^{-1})(x)+C$$ Which gives: $$I=x\arcsin x-(-\cos(\arcsin x))+C$$ $$I=x\arcsin x+\cos\arcsin x+C$$ Note the following: $$\alpha=\arcsin x$$ $$\therefore \sin\alpha=x$$ Recall and apply the Pythagorean identity: $\cos^2\alpha=1-\sin^2\alpha$ $$\therefore \cos\alpha=\sqrt{1-x^2}$$ Plug it in: $$I=x\arcsin x+\sqrt{1-x^2}+C$$ You got it!