Choosing The Inner Function

Why the Choice of Inner Function Matters

A substitution can simplify an integral by undoing the chain rule. The main decision is often not how to integrate the transformed expression, but which part of the original integrand to call \(u\).

When an integrand contains a nested expression, such as a power of a polynomial or a radical involving a polynomial, a useful first question is: “If I choose this nested expression as \(u\), does its derivative already appear elsewhere in the integrand?” When the answer is yes, the substitution is usually direct.

This is a selection principle, not a claim that every integral has an obvious substitution. A candidate is useful when replacing it with \(u\) makes the whole integrand simpler and accounts for the accompanying factors.

The Matching Principle

Suppose an integrand has the form \(f(g(x))g'(x)\), where \(f\) and \(g\) are differentiable on the interval being considered. Set \(u=g(x)\). The substitution rule gives

The function \(g(x)\) is the inner function: it is the expression inside another function. For instance, in \((2x^2+3)^7\), the expression \(2x^2+3\) is nested inside the seventh power. Its derivative is \(4x\), so an integrand containing \(4x(2x^2+3)^7\) has a direct match.

In practice, inspect the integrand for nested expressions, differentiate promising candidates, and compare those derivatives with the other factors. The derivative need not be written next to the candidate; it may be a factor elsewhere in the product. If it is absent, the candidate may not simplify the integral by direct substitution.

Worked Example: A Power of a Quadratic

Worked Example: A Radical

Worked Example: A Rational Expression

Why the Matching Principle Works

The key justification is the chain rule. If \(F\) is an antiderivative of \(f\), so \(F'(u)=f(u)\), then differentiating the composition \(F(g(x))\) gives

Thus, whenever the integrand is \(f(g(x))g'(x)\), the composition \(F(g(x))\) differentiates to that integrand. Substitution expresses this reversal in a useful way: replace \(g(x)\) by \(u\), integrate the resulting expression in \(u\), then replace \(u\) by \(g(x)\). The derivative match is not a coincidence; it is precisely the factor created by the chain rule.

Choosing Among Possible Candidates

Sometimes several nested expressions seem possible. Prefer the candidate whose derivative is already present and whose substitution leaves the simplest expression. In \(\int x(1+x^2)^4\,dx\), the natural candidate is \(u=1+x^2\), because its derivative is \(2x\), closely matching the factor \(x\), while the remaining power becomes \(u^4\). Choosing the full expression \((1+x^2)^4\) instead introduces a derivative containing the extra factor \((1+x^2)^3\), so the transformed expression is less direct.

A candidate does not become useful merely because it is nested. After substituting, check the whole integrand: the remaining factors must also be expressible using \(u\) and its differential. If an \(x\)-factor remains that cannot be rewritten simply in terms of \(u\), the proposed choice may not help.

For this tutorial’s basic matching cases, the clearest choice is one whose derivative appears exactly as a factor. More complicated situations can require rewriting an expression or accounting for a constant factor; those require additional care. The guiding question remains the same: does the choice turn the original integral into a simpler one without losing any factors?

A Common Mistake: Ignoring the Derivative Factor

Consider the tempting calculation

This treats \(3x^2+1\) as though its derivative were already present. But the proposed answer differentiates to

which is not the integrand \((3x^2+1)^5\). The substitution \(u=3x^2+1\) produces \(du=6x\,dx\), but the original integral has no factor \(6x\). Therefore this direct substitution does not turn the integral into a power of \(u\) alone. The correction is not to ignore the missing factor: first check whether the derivative of the chosen inner function is actually available in the integrand.

By contrast, for \(\int 6x(3x^2+1)^5\,dx\), the factor \(6x\) is present. The same substitution gives \(\int u^5\,du\), and therefore

Differentiating this expression gives \(6x(3x^2+1)^5\), exactly as required. The difference between the two integrals is the derivative factor.

Using the Choice as a Reliable Habit

Choosing \(u\) is a way to organize what you see, not a guessing game. Look for a function inside another function; differentiate that candidate; then compare its derivative with the remaining factors. Powers, radicals, and denominators are common places to find a useful inner expression, but the derivative match is what makes the selection work.

Before accepting an answer, verify two things: the substitution accounted for every factor, and differentiating the final expression returns the original integrand. This check is especially helpful when an expression looks familiar, since a familiar outer power does not by itself justify treating the inner expression as a variable.