I have discovered this general proof of rule of substitution that I came across while taking my course in integral calculus. I was bothered by this “multiplying” $dx$ thing but did not find a satisfactory explanation either on the Internet or in books. So I proved it myself.
It is general in the sense that it does not deal with $t=g(x)$ or $g(x)=t$ cases separately. In these cases, invertibility of function also comes into picture.