Why are variables in integration by substitution so counter intuitive?

Question

Integration by substitution is defined as something like

$\displaystyle\int_a^b f(\phi(t))\phi'(t)dt = \int_{\phi(a)}^{\phi(b)} f(x)dx$

But for my taste, the variables $x$ and $t$ are exactly switched! It's still correct since it's just a name, but why is this written in the exact counter intuitive way? I Feel like maybe I'm missing something. To illustrate, an example:

$\displaystyle\int_{x=0}^{x=2} x \cos(x^2+1)dx$

Then we have $t=ϕ(x)=x^2+1$ and therefore $\frac{dt}{dx}=2x$ or $dx=\frac{dt}{2x}$. So we have

$\displaystyle\int_{\phi(0)}^{\phi(2)} \frac 1 2 \cos(t)dt = \int_{t=1}^{t=5} \frac 1 2 \cos(t)dt = \frac 1 2 (\sin 5 - \sin 1)$

Which is all correct and wonderful. But why on earth would you name the variables $t$ and $x$ exactly the opposite way of how they would appear in almost every situation?

Answer 1

Since forever we have studied functions with the idea of the variable $x$ and write $y=f(x)$ for the graph of $f$ in the coordinate plane. The next common letter for a dummy variable is $t$, possibly from physic for time.

Similarly, when study integration of a function, it's common that we write the integration as $\int f(x)dx$. And most formulas/exercises are given in this form (power law, exponential law,...). It's up to the writer's taste to choose the notations.

In this case my guess is that when he wrote the Substitution Law, he had the picture of the final integral in mind. That is, the last integral that the students/readers would encounter in the computation process.

I would like to add that many other books choose different notations. For example $$\int^b_a f(u(x))u'(x)dx= \int^{u(b)}_{u(a)}f(u)du.$$ This choice puts the original integral first which in some cases helps the students recognize the substitution patterns faster. Again, it's totally personal taste and I wouldn't take that too seriously.

Answer 2

If $\int_a^b f(x)dx$, then you set $x=g(y)$ then if $x$ move from $a\to b$ then $y$ move from $g^{-1}(a)\to g^{-1}(b)$ and $dx=dg(y)= \frac{dg(y)}{dy}\cdot dy=g'(y)dy$ then $$\int_a^b f(x)dx=\int_{g^{-1}(a)}^{g^{-1}(b)}f(g(y))g'(y)dy$$

I hope it answer at your question.

Answer 3

The variables $x$ and $t$ are dummy variables. This means, essentially, that we can rename $x$ and $t$ without changing anything of substance. You might as well write the change of variables formula as $$ \int_a^b f(\phi(x))\,\phi'(x)\,dx = \int_{\phi(a)}^{\phi(b)} f(t)\,dt. $$ (Does this answer your question? I'm not sure that I know what you're asking.)