Why Rewrite the Numerator?
A rational integrand can be difficult to recognize when its numerator does not immediately resemble the derivative of its denominator. A useful first step is to rearrange the numerator: separate out a multiple of the denominator’s derivative, then see what remains.
This can expose a logarithmic term. If the leftover part also simplifies—for example, because it is a multiple of the denominator—it may produce an elementary term as well. The rearrangement does not guarantee that every integral becomes easy, so it is important to inspect the remainder rather than discard it.
The Derivative-Matching Identity
Suppose \(Q\) is differentiable and \(Q(x)\ne 0\) on an interval. The derivative of \(\ln|Q(x)|\) is \(Q'(x)/Q(x)\). Consequently, if a numerator \(P(x)\) can be written as \(cQ'(x)+R(x)\), then the fraction splits into a logarithmic-pattern term and a remainder term.
The split is useful when the second fraction is also straightforward to integrate. One especially convenient case is \(R(x)=S(x)Q(x)\), where \(S\) is a polynomial. Then the denominator cancels from the remainder term, leaving just \(S(x)\).
The absolute value is needed because the real logarithm \(\ln|Q(x)|\) is defined when \(Q(x)\) is either positive or negative, as long as it is nonzero. An antiderivative is understood on an interval that does not include a zero of the denominator.
Why the Rewrite Works
The identity follows from the derivative rule for a logarithm and the linearity of integration. Here is the full justification. Assume \(Q(x)\ne 0\) on the interval under consideration and \(P(x)=cQ'(x)+S(x)Q(x)\). Then
By the chain rule, differentiating \(\ln|Q(x)|\) gives \(Q'(x)/Q(x)\) wherever \(Q(x)\ne 0\). Therefore
Thus the proposed expression is an antiderivative. This explanation also shows why the match matters: the logarithmic derivative has precisely the form “derivative of a function divided by that function.”
A Practical Way to Arrange the Numerator
For a polynomial denominator \(Q(x)\), first calculate \(Q'(x)\). Compare its terms and coefficients with the numerator. If a constant multiple of \(Q'(x)\) accounts for part of the numerator, subtract that part to find the remainder. Then divide the remainder by \(Q(x)\), or recognize it directly as \(S(x)Q(x)\), if possible.
For a denominator \(Q(x)\), compute \(Q'(x)\) and identify terms that could match the numerator.
Write it as a multiple of \(Q'(x)\) plus the exact remainder. Check the equality by adding the pieces back together.
The derivative-matching part gives a logarithm. Determine whether the remainder divided by \(Q(x)\) simplifies to an elementary expression.
Include the constant of integration, then differentiate the result to recover the original integrand.
Worked Examples
Worked Example: A Logarithm Plus a Polynomial Term
Evaluate \(\displaystyle \int \frac{4x+2x^2+6}{x^2+3}\,dx\).
Set \(Q(x)=x^2+3\). Its derivative is \(Q'(x)=2x\). Rewrite the numerator using both \(Q'\) and \(Q\):
Substitute this split into the integrand and simplify each term:
Since \(x^2+3>0\) for every real \(x\), the absolute value could be omitted in this particular answer. Keeping it makes the logarithmic pattern explicit. Check by differentiating:
The derivative matches the original integrand.
Worked Example: A Polynomial Remainder
Evaluate \(\displaystyle \int \frac{3x^3-2x^2+9x-2}{x^2+1}\,dx\).
Let \(Q(x)=x^2+1\), so \(Q'(x)=2x\). The numerator can be written as a multiple of \(Q'\) plus a polynomial multiple of \(Q\):
Expanding the right-hand side verifies the split: \(6x+(3x^3-2x^2+3x-2)=3x^3-2x^2+9x-2\). Therefore
Because \(x^2+1>0\), this logarithm could also be written without absolute value bars. To check, differentiate the proposed antiderivative:
This recovers the original integrand.
Worked Example: Substitution with a Squared Denominator
Evaluate \(\displaystyle \int \frac{4x^3+8x}{(x^2+1)^2}\,dx\).
Here the useful substitution is \(u=x^2+1\), so \(du=2x\,dx\). Rewrite the numerator to expose \(2x\,dx\), and express the remaining polynomial factor in terms of \(u\):
Since \((x^2+1)^2=u^2\), the integral becomes
For the check, differentiate the answer. Write \(u=x^2+1\), so \(u'=2x\):
The derivative agrees with the integrand, so the antiderivative is correct.
When the Remainder Is Not Finished
A derivative match is a useful start, not a guarantee that the whole integral is solved. For example, after writing \(P(x)=cQ'(x)+R(x)\), the remaining integral is \(\int R(x)/Q(x)\,dx\). If that term does not simplify to a known elementary integral, it still requires a method of its own. Do not label the original integral complete just because its first part became a logarithm.
For instance, a constant remainder over a quadratic denominator may leave an integral that is not a polynomial term and is not the logarithmic derivative pattern. The split is still algebraically correct, but more work is needed to evaluate the remainder. The present strategy is most direct when the remainder is zero, cancels with the denominator, or becomes a simple expression after substitution.
Common Mistake: Keeping Only the Logarithm
In the first worked example, the numerator is \(2Q'(x)+2Q(x)\). A common error is to notice \(2Q'(x)\), integrate only that piece, and write \(2\ln|Q(x)|+C\). This drops the \(2Q(x)\) part of the numerator. The wrong check makes the problem visible:
The correction is to divide the full numerator split by the denominator:
The second term integrates to \(2x\), so the complete answer is \(2\ln|x^2+3|+2x+C\). The reliable habit is to verify that the split adds back to the entire original numerator before integrating.
Summary
When a rational integrand has a polynomial numerator and denominator, differentiate the denominator and look for a matching part in the numerator. Write the numerator as a derivative-matching term plus its exact remainder. The matching term gives a logarithm; the remainder must also be simplified and integrated. If substitution is used, express every remaining factor in the new variable and transform the differential consistently.
Check Your Understanding
Answer these questions, and check any proposed antiderivative by differentiating.
- For \(Q(x)=x^2+5\), what is \(Q'(x)\), and what numerator term produces a logarithmic derivative over \(Q(x)\)?
- Rewrite \(2x+3(x^2+1)\) as a derivative-matching term plus a multiple of \(Q(x)=x^2+1\).
- Evaluate \(\displaystyle \int \frac{2x+4(x^2+1)}{x^2+1}\,dx\).
- Why is \(\ln|Q(x)|\), rather than always \(\ln(Q(x))\), used in the general formula?
- If the derivative-matching split leaves \(R(x)/Q(x)\), what must you check before claiming the integral is complete?
