A Minor Caution with the Chain Rule

When we go about to commit an act of integration, the chain rule tells us that we can replace the variable in our integrand with a suitable function of some other variable. Thus when we want to integrate

(Eq'n 1)

we can make the substitution x = g(y) and dx = (dg/dy)dy, so that we have

(Eq'n 2)

For example, we could take the integral

(Eq'n 3)

and replace x with y^{n} and make dx = ny^{n-1}dy.
The integral then becomes

(Eq'n 4)

as we expect.

Aside from such trivial applications, which I used previously to refine our knowledge of the applicability of Equation 3, to extend the domain of its exponent to the realm of fractions, we want to use the chain rule to make opaque integrands more transparent, to enable us to solve their integrals more readily. But the chain rule does not grant us unconstrained license to employ any substitution that we find convenient. As an example of how misapplication of the chain rule can lead us astray, consider the integral

(Eq'n 5)

Let's make the substitution x = yz so that dx = ydz + zdy. Then we have

(Eq'n 6)

which is wrong.

In this example I clearly went wrong by replacing the single
independent variable with the product of two independent variables. From that
lone fact we may infer that if we want to guarantee the correctness of our
solution of an integral, then we cannot, in the course of applying the chain
rule, change the number of independent variables in the integrand. As we strive
to devise ever more clever ploys to solve ever more difficult integrals, such
rules will become indispensable guides to the solutions that are always true to
mathematics.

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