Introduction
A rational expression can become much easier to integrate when its numerator contains the derivative of the expression being used inside the denominator. The key is to notice the relationship between the composite expression and its derivative, not merely the fact that the integrand contains a fraction.
For instance, in an expression involving \(x^2+1\), the derivative of that inner expression is \(2x\). If the integrand contains a factor of \(2x\), the substitution \(u=x^2+1\) may replace the entire composite structure with an ordinary rational function of \(u\). If that factor is missing, the same substitution may introduce a square root and make the integral less convenient.
The Structure of a Suitable Rational Composite
A rational function of \(u\) is a quotient \(R(u)=P(u)/Q(u)\), where \(P\) and \(Q\) are polynomials and \(Q(u)\ne 0\) wherever the expression is being considered. In this tutorial, the useful pattern is an integrand of the form \(R(g(x))g'(x)\): a rational function evaluated at a composite expression, multiplied by the derivative of that expression.
The phrase “suited to substitution” does not mean that the resulting integral is automatically immediate. It means the composite expression and its derivative fit together, so the substitution removes the composition cleanly. The resulting rational integral may still require algebraic simplification or a decomposition into simpler fractions.
For an indefinite integral, the underlying justification is the chain rule. If \(H'(u)=R(u)\) on an interval containing the values of \(g(x)\), then
Thus \(H(g(x))\) is an antiderivative of the original integrand on any interval where the expressions are defined. This argument does not require \(g\) to be one-to-one. It does require the relevant denominator not to vanish on the interval, and it requires the inner function to be differentiable there.
Worked Example: A Power of a Composite Denominator
Worked Example: A Power of a Composite Denominator
Evaluate \(\int \frac{6x}{(x^2+4)^3}\,dx\).
The repeated expression in the denominator is \(x^2+4\). Its derivative is \(2x\), and the numerator \(6x\) is three times that derivative. Set \(u=x^2+4\). Then
Substitution gives
Using the power rule,
Returning to \(x\), an antiderivative is \(-\frac{3}{2(x^2+4)^2}+C\). To check, differentiate it:
This is the original integrand, so the calculation is correct.
Worked Example: Substitution Followed by Partial Fractions
Worked Example: Substitution Followed by Partial Fractions
Evaluate \(\int \frac{2x}{(x^2+1)(x^2+2)}\,dx\).
Take \(u=x^2+1\). Then \(du=2x\,dx\), and \(x^2+2=u+1\). Both quadratic factors become simple expressions in \(u\):
The substitution has produced a rational function, but it is not yet in a form that can be integrated by a single basic rule. Decompose it into partial fractions. Since
we obtain
For real \(x\), \(u=x^2+1>0\) and \(u+1=x^2+2>0\), so the absolute values can be removed after returning to \(x\):
To check, differentiate this expression:
The result matches the original integrand.
Why the Substitution Works
The preceding examples use the same principle, even though the rational integrals after substitution differ. Here is the full chain-rule justification in general form.
By assumption, \(H'(u)=R(u)\) throughout the interval containing the values of \(g(x)\).
The function \(H(g(x))\) is defined on \(I\), and the chain rule gives \(\frac{d}{dx}H(g(x))=H'(g(x))g'(x)\).
Substituting \(H'(g(x))=R(g(x))\) yields \(\frac{d}{dx}H(g(x))=R(g(x))g'(x)\), the original integrand.
This proves the result: \(H(g(x))\) is an antiderivative. In substitution notation, \(u=g(x)\) and \(du=g'(x)\,dx\) are a compact way to carry out this chain-rule reasoning. The notation is useful only when the factors really match.
When the Composite Derivative Is Missing
A denominator that contains a composite expression is not, by itself, enough to make substitution helpful. Compare \(\frac{1}{(x^2+1)^2}\) with the integrand in the first example. The inner expression is again \(x^2+1\), but its derivative \(2x\) is absent. Setting \(u=x^2+1\) does not simply turn the integral into a rational function of \(u\).
Worked Example: A Substitution That Does Not Simplify the Integral
Consider \(\int \frac{1}{(x^2+1)^2}\,dx\) and test the proposed substitution \(u=x^2+1\) on the interval \(x>0\). The restriction makes \(x=\sqrt{u-1}\), and
Therefore this substitution transforms the integral into
The transformed expression still contains a square root, so the substitution has not converted the integral into a rational integral in \(u\). This does not show that the original integral has no antiderivative or that it cannot be evaluated by another method. It shows only that this proposed substitution has not produced the intended simplification.
A Common Mistake: Replacing \(dx\) Incorrectly
When \(u=x^2+1\), the equation \(du=2x\,dx\) does not imply \(dx=du/2\). Dividing by \(2\) alone drops the factor \(x\). The correct rearrangement, where \(x\ne 0\), is \(dx=du/(2x)\); if one then expresses everything in \(u\), \(x\) must also be rewritten in terms of \(u\) on a suitable interval.
A related diagnostic is to look for a constant multiple of \(g'(x)\), not only an exact match. A constant can be factored out, as in the first example. But a factor depending on \(x\) cannot be discarded. If the derivative factor is only partly present, first consider whether the integrand can be split or algebraically rewritten; do not assume that the whole integral has the desired form.
A Practical Test for Rational Composite Substitution
Before substituting, identify the candidate inner expression and compare its derivative with the rest of the integrand. After substituting, check what the new integrand actually contains. A successful substitution should remove the composite expression from its nested role and leave a rational function of the new variable, possibly ready for further algebra.
For example, in a denominator involving \(x^2+1\), test \(u=x^2+1\).
Compute \(du=g'(x)\,dx\) explicitly, including any constant factor.
Check whether the required \(g'(x)\) factor is present, or whether the integrand can be rewritten to expose it.
Rewrite every occurrence of the original variable that remains. If this introduces extra factors such as square roots and leaves the expression more complicated, the substitution is not simplifying the problem.
Evaluate the resulting rational integral, substitute back, and differentiate the proposed antiderivative to check it.
Why This Recognition Matters
Rational composites appear in many forms: a power of a polynomial expression, a product of factors built from the same inner expression, or a quotient whose numerator matches the derivative of its denominator’s inner expression. Recognizing the structure can reduce a complicated-looking integral to a familiar one-variable rational integral. The substitution does not make the algebra disappear; it isolates it in the right variable.
The reverse lesson matters just as much. A visually prominent denominator is not a sufficient reason to choose its inner expression as a substitution. The derivative factor is the evidence that the change of variable fits. Without it, the substitution may require solving for \(x\) and inserting additional factors, as in the example with \(x^2+1\). A reliable method therefore combines pattern recognition with an explicit check of \(du\).
Check Your Understanding
For each question, focus on the relationship between the proposed inner expression and its derivative.
- For \(\int \frac{4x}{(x^2+3)^2}\,dx\), identify a suitable substitution and write the transformed integral before evaluating it.
- Evaluate \(\int \frac{3x^2}{x^3+5}\,dx\), and verify the answer by differentiation.
- For \(\int \frac{dx}{(x^2+4)^3}\), explain why \(u=x^2+4\) does not immediately produce a rational integral in \(u\).
- State the chain-rule condition that justifies integrating \(R(g(x))g'(x)\) by first finding an antiderivative of \(R(u)\).
- In a proposed substitution, what must be checked before replacing \(dx\) using the relation between \(du\) and \(dx\)?
