Why a Plausible Substitution Can Fail
A substitution can look promising because an integrand contains a familiar inner expression, such as \(x^2+1\) or \(x^2+2x+5\). But spotting an inner expression is only the first check. The integrand must also contain its derivative, or a constant multiple of that derivative, for the reverse chain rule to apply directly.
This is the missing-factor trap: integrating a function of an inner expression as if the inner derivative were present, even when it is absent. A careful match prevents incorrect answers and helps distinguish a direct substitution from one that requires algebraic rewriting or a different method.
The Reverse Chain Rule
The chain rule says that differentiating a composition \(F(g(x))\) produces a factor of \(g'(x)\). Reversing that pattern gives a basic integration rule. Suppose \(F'(u)=f(u)\) on an interval and \(g\) is differentiable. Then
Consequently, whenever the integrand has the form \(f(g(x))g'(x)\), an antiderivative is \(F(g(x))\). A constant factor is also permitted: if the integrand is \(c f(g(x))g'(x)\), its antiderivative is \(cF(g(x))\), where \(c\) is constant.
In substitution notation, we often set \(u=g(x)\), so \(du=g'(x)\,dx\). This notation is a compact way of recording the same derivative match. It does not make a missing factor appear: if the integrand has no \(g'(x)\), the substitution does not automatically produce an integral of \(f(u)\).
Why the Rule Works
The justification is the chain rule itself. Suppose \(F'=f\), and consider the proposed antiderivative \(F(g(x))\). Differentiating it gives
This derivative is exactly the integrand when the required factor is present. If the integrand is \(c f(g(x))g'(x)\), differentiating \(cF(g(x))\) gives \(c f(g(x))g'(x)\), since \(c\) is constant. Thus the match is not based on visual similarity alone; it is verified by differentiation.
Identify the expression \(g(x)\) that appears inside a power, logarithm, trigonometric function, or other outer function.
Calculate \(g'(x)\) explicitly rather than relying on a visual guess.
Check whether \(g'(x)\) is present as a factor, possibly after taking out a constant or rearranging terms.
Apply the corresponding outer antiderivative, then differentiate the result to check it.
Exact Matches and Constant-Factor Matches
An exact match contains the inner derivative as written. A constant-factor match contains a fixed multiple of it. In the second case, the constant must be accounted for; it cannot simply be ignored.
Worked Example: An Exact Inner-Derivative Match
Evaluate \(\int 2x\cos(x^2+1)\,dx\).
The inner expression is \(g(x)=x^2+1\), so \(g'(x)=2x\). The integrand contains precisely this factor:
An antiderivative of \(\cos u\) is \(\sin u\). Therefore, the reverse chain rule gives
Check by differentiation:
This is the original integrand, so the answer checks.
Worked Example: Restoring a Missing Constant Factor
Evaluate \(\int \frac{x}{x^2+4}\,dx\).
Choose \(g(x)=x^2+4\). Then \(g'(x)=2x\), while the numerator is \(x\), half of \(g'(x)\). Rewrite the numerator to display that factor:
Hence
The derivative of \(\ln|g(x)|\) is \(g'(x)/g(x)\) wherever \(g(x)\ne 0\). Here \(x^2+4>0\) for every real \(x\), so the absolute-value signs can be omitted:
Check by differentiating:
The factor \(\frac12\) is essential: it compensates for the fact that the numerator is half the inner derivative.
When Rewriting Reveals Only Part of a Match
Sometimes the numerator is not just a constant multiple of the denominator's derivative. It may be possible to split it into a matching part and a remainder. The reverse chain rule then integrates the matching part, but it does not automatically handle the remainder. That remaining integral still needs to be evaluated by an appropriate method.
Worked Example: Splitting Off the Derivative Part
Evaluate \(\int \frac{x+3}{x^2+2x+5}\,dx\).
Let \(g(x)=x^2+2x+5\). Its derivative is \(g'(x)=2x+2\). The numerator is not a constant multiple of \(2x+2\), but it can be rewritten as
Separate the integral using this identity:
The first integral is a direct reverse-chain-rule match. For the second, complete the square:
Using \(\int \frac{1}{z^2+a^2}\,dz=\frac1a\arctan(z/a)+C\), with \(z=x+1\) and \(a=2\), gives
The logarithm is valid without absolute-value signs because \(x^2+2x+5=(x+1)^2+4>0\). Check the proposed answer term by term:
Adding these derivatives gives
which is the original integrand. The rewrite exposed a derivative match for one part, not for the entire numerator.
The Missing-Factor Trap
A common error is to identify an inner expression and then integrate its outer function without checking the inner derivative. For example, the fact that \(x^2\) appears inside a cosine does not mean that \(\int \cos(x^2)\,dx\) is obtained by simply integrating \(\cos u\) with \(u=x^2\).
Worked Example: Rejecting an Invalid Shortcut
Consider the attempted calculation
To check it, differentiate the proposed answer:
This is not \(\cos(x^2)\); an extra factor \(x\) appears. The proposed inner function \(g(x)=x^2\) has derivative \(2x\), and the original integrand contains no factor proportional to \(x\). On the interval \(x>0\), the substitution \(u=x^2\) instead gives \(du=2x\,dx\), so \(dx=\frac{du}{2\sqrt{u}}\). The transformed integral is
This is not the simple integral \(\int\cos u\,du\). The substitution has not produced the claimed sine antiderivative. The derivative check exposes the mistake immediately.
What to Do When the Factor Is Absent
If the proposed inner derivative is absent, do not insert it by assumption. First check whether ordinary algebra can reveal a matching component, as in the numerator split above. If no useful rewrite exists, the proposed substitution may simply be a poor choice for that integrand. A substitution is a tool, not a requirement: it is useful when the transformed integral is valid and manageable.
For logarithmic forms, a particularly common match is \(\frac{g'(x)}{g(x)}\). The precise rule is
on any interval where \(g\) is differentiable and nonzero. If the numerator is \(c\,g'(x)\), the answer is \(c\ln|g(x)|+C\). If the numerator differs in a nonconstant way, it is not enough to label the denominator as \(u\); the unmatched part must be dealt with separately.
A Reliable Check Before Integrating
Before using reverse chain rule, write down the proposed inner function and its derivative. Then compare the derivative with the actual factors in the integrand. Keep track of any constant multiplier, and check whether algebraic rewriting splits the expression into more than one term. Finally, differentiate the proposed antiderivative. That last step tests the result against the original integrand rather than against the substitution you hoped would work.
Check Your Understanding
For each question, identify the proposed inner function and decide whether the derivative match is exact, differs by a constant factor, can be exposed by rewriting, or is absent.
- For \(\int 6x(3x^2+1)^4\,dx\), what is the inner function, and what makes this a direct match?
- Evaluate \(\int \frac{3x^2}{x^3+2}\,dx\). What constant factor, if any, must be included?
- In \(\int \frac{x+4}{x^2+4x+6}\,dx\), rewrite the numerator using the derivative of the denominator. Does this produce only a derivative-matched term?
- Why does the proposed antiderivative \(\frac12\sin(x^2)+C\) fail for \(\int\cos(x^2)\,dx\)?
- State the condition on \(g\) that allows \(\int \frac{g'(x)}{g(x)}\,dx=\ln|g(x)|+C\).
