Why Rewrite Before Substituting?
In some integrals, the expression to substitute is easy to recognize, but the rest of the integrand does not yet look like its derivative. A small algebraic rewrite can make the relationship visible. The goal is not to change the value of the integrand; it is to express it in a form where the reverse chain rule can be applied.
For example, if an integrand contains a power of \(x^2+4\), it may be natural to consider \(u=x^2+4\), whose differential is \(du=2x\,dx\). If the integrand contains \(x^3\,dx\), the factor \(x\) is present, but the remaining \(x^2\) can first be rewritten as \((x^2+4)-4\). That is the key kind of algebraic preparation studied here.
The Algebra Behind Substitution
The substitution method is a form of the reverse chain rule. If \(u=g(x)\), then \(du=g'(x)\,dx\). An integral is especially direct when its integrand can be written as \(f(g(x))g'(x)\), or as a constant multiple of that form. The algebraic rewrite happens before changing variables: it helps show that the needed factors are actually present.
When the derivative factor is present only up to a constant, account for that constant explicitly. For instance, if \(u=2x^2+6x+7\), then \(du=(4x+6)\,dx\). If the integrand has \(4x+6\), there is no extra factor to supply. If it has \(2x+3\), then \(du=2(2x+3)\,dx\), so the substitution introduces a factor of \(1/2\).
Sometimes only part of the integrand fits a substitution. In that case, an identity may split the integrand into a sum of simpler terms. Each term must still be integrated, and a substitution should be used only where the corresponding differential factor is present.
Worked Example: Match a Numerator to a Denominator's Derivative
Worked Example: Match a Numerator to a Denominator's Derivative
Evaluate \(\displaystyle \int \frac{4x+6}{2x^2+6x+7}\,dx\).
Let the denominator be the inner expression:
Differentiate it:
The numerator and \(dx\) are exactly \(du\), so the integral becomes
Substitute back. The quadratic is positive for every real \(x\): its leading coefficient is positive and its discriminant is \(6^2-4(2)(7)=-20<0\). Thus the absolute value can be omitted:
Check: Differentiating the proposed antiderivative gives
which is the original integrand.
This example illustrates a useful pattern: a quotient with a denominator \(g(x)\) and numerator \(g'(x)\) integrates to \(\ln|g(x)|+C\). If the numerator is a constant multiple of \(g'(x)\), that constant must be included in the answer. The algebraic task is to compare the numerator with the derivative of the entire denominator, not just with one term of it.
Rewrite Powers Using the Inner Expression
A common obstacle is having too many powers of \(x\), or the wrong-looking power, alongside a composite expression. If the inner expression is \(u=x^2+a\), then \(x^2=u-a\). This identity lets you rewrite powers of \(x\) in terms of \(u\), while keeping the factor \(x\,dx\) for the differential.
Worked Example: Rewrite a Power Before Substituting
Evaluate \(\displaystyle \int x^3\sqrt{x^2+4}\,dx\).
The radical suggests \(u=x^2+4\), so \(du=2x\,dx\), or \(x\,dx=\frac12du\). The integrand has \(x^3\,dx=x^2(x\,dx)\). Rewrite \(x^2\) using the proposed inner expression:
Therefore the entire integrand can be rearranged as
Now set \(u=x^2+4\). Since \(\sqrt{x^2+4}=u^{1/2}\), substitution gives
Integrate each power:
Substituting back gives
Check: Differentiate the result using the chain rule:
The last line is exactly the original integrand.
Split the Integrand When That Makes the Pattern Clear
An algebraic identity can also separate an integrand into a term that is easy to integrate directly and another term suited to substitution. The identity must preserve the whole expression, including any factors outside the fraction. A reliable check is to combine the rewritten terms again and confirm that the original integrand is recovered.
Worked Example: Divide and Split Before Substitution
Evaluate \(\displaystyle \int \frac{x(x^2+3)}{x^2+1}\,dx\).
The numerator has a factor \(x\), which is useful for \(u=x^2+1\), since \(du=2x\,dx\). But the factor \(x^2+3\) is not simply the denominator. Rewrite it as
Then split the fraction:
Thus
The first term has antiderivative \(x^2/2\). For the second, let \(u=x^2+1\), so \(du=2x\,dx\):
Combining the terms, an antiderivative is
Check:
This matches the original integrand.
Why the Rewrite Works
The substitution itself works because of the chain rule. If \(G\) is an antiderivative of \(f\), then differentiating the composite function \(G(g(x))\) gives
Consequently, whenever the integrand can be rewritten as \(f(g(x))g'(x)\), the function \(G(g(x))\) is an antiderivative. A constant multiple is handled by keeping that constant outside the integral. The algebraic rewrite does not create a new calculus rule; it reveals the factors needed to use the reverse chain rule.
For instance, in the second example the identity \(x^2=(x^2+4)-4\) turned \(x^3\,dx\) into a sum of terms involving \(x\,dx\) multiplied by powers of \(x^2+4\). That made the differential \(du=2x\,dx\) available in every term. The substitution then converted the remaining factors into powers of \(u\).
A Common Mistake: Replacing Only Part of a Power
In the radical example, a tempting but incorrect step is to write \(x^3=xu\) after setting \(u=x^2+4\). This drops the \(4\), because \(u\) is not equal to \(x^2\). The incorrect calculation would give
The correction is to solve the substitution equation for the power being replaced:
Then use \(x\,dx=\frac12du\). Keeping the constant term in this identity is essential; it changes the integral and its answer. More generally, write the full identity first, and check it by expanding or substituting back before proceeding.
A Practical Algebra Check
Before integrating, verify that the rewrite is an identity and that the differential is accounted for. These checks help distinguish a useful rearrangement from an accidental change to the integrand.
- Check the inner expression: Write down \(u=g(x)\) and compute \(du=g'(x)\,dx\).
- Check the algebra: Expand, factor, or recombine the rewritten terms to recover the original integrand.
- Check constant factors: If \(du\) is twice or half the factor in the integral, carry that multiplier through the substitution.
- Check the result: Differentiate the proposed antiderivative and compare it with the original integrand.
These checks are particularly useful when an expression has several terms or powers. Substitution is not a reason to discard a factor that does not appear to fit; algebra must account for every part of the original integrand.
Check Your Understanding
For each question, focus on the algebraic identity that makes the substitution possible.
- For \(\displaystyle \int \frac{6x+2}{3x^2+2x+5}\,dx\), identify an inner expression and its differential.
- When \(u=x^2+9\), rewrite \(x^4\) in terms of \(u\).
- Rewrite \(\displaystyle \frac{x(x^2+5)}{x^2+1}\) as a sum in which one term is a constant multiple of \(\frac{x}{x^2+1}\).
- Why is \(x^2=u-4\), rather than \(x^2=u\), when \(u=x^2+4\)?
- What calculation can confirm that a proposed antiderivative is correct?
