Change of variables in integrals

It is important when dealing with change of variables to always remember what an integration of a function means:

\[ \int_{x_0}^{x_1} f(x)\,\,dx \]

Recall we are summing rectangles of base \(dx\) and height \(f(x)\) with \(x\) ranging between \(x_0\) and \(x_1\).

In other words, the integral means, for each \(x\) in this range compute the area \(f(x)dx\) and add the final result.

Note that for each \(x\) there is a corresponding \(f(x)dx\); changing the variable \(x\) to some new variable \(y\) must not change this assignment.

Assume \(x\) is related to \(y\) by some function \(g\), then \(x=g(y)\). The new variable \(y\) must now have a new range and this range must be in such a way that the corresponding \(g(y)\) ranges between \(x_0\) and \(x_1\) without repeating values. That is to say the function \(g\), must be bijective!

If this is the case, then, the variable \(y\) now ranges between \([g^{-1}(x_0) , g^{-1}(x_1)]\) and each \(dx\) (we have one \(dx\) for each \(y\)) is obtained from the linear approximation

\[ dx=g'(y)dy \]

In conclusion:

\[ \int_{x_0}^{x_1} f(x)\,\,dx = \int_{g^{-1}(x_0)}^{g^{-1}(x_1)} f(g(y))g'(y)\,\,dy \]

Hopefully you choice of \(g(y)\) make the integral on the rhs easier to evaluate.