Why Mixed Integrands Need a Plan
An integrand may contain several recognizable features at once: a product, a rational expression, a trigonometric function, or a composite function. Spotting a familiar pattern is useful, but it does not automatically tell you which method to apply first. A small algebraic rewrite can make the right method much clearer.
The goal is to arrange the work so that each step prepares the next one. For example, simplify a rational expression before trying substitution, or split a numerator into terms that match different standard antiderivatives. The aim is not to use as many methods as possible. It is to use only the methods that make the integral manageable.
A Practical Order of Decisions
Begin by checking whether the expression can be simplified without changing its value on the domain of the integrand. Look for factoring, expansion, cancellation, or division of polynomials. Next, check whether a numerator or factor is the derivative of an expression appearing inside a composite function. If so, substitution may be appropriate. Trigonometric identities can also reveal a derivative pairing or reduce powers. If a product remains and one factor becomes simpler when differentiated while the other has a manageable antiderivative, integration by parts may be a candidate.
| Feature to inspect | Useful first move |
|---|---|
| A rational expression with numerator degree at least the denominator degree | Consider polynomial division or an algebraic rewrite. |
| A function of \(g(x)\) multiplied by a factor involving \(g'(x)\) | Test the substitution \(u=g(x)\). |
| A trigonometric expression with a recognizable identity or derivative pairing | Rewrite it using an appropriate trigonometric identity. |
| A product that remains difficult after simplification | Check whether integration by parts is a candidate. |
| A denominator that resembles a standard form after completing the square | Rewrite the quadratic before choosing a standard antiderivative. |
These are prompts, not automatic rules. A product does not always call for integration by parts, and a visible composite function does not guarantee that substitution will help. The proposed first move should make the remaining integrand simpler or better matched to a known form.
Worked Example: Divide Before Integrating
Worked Example: Simplifying a Rational Integrand First
Evaluate \(\int \frac{x^3}{x^2+1}\,dx\).
The numerator has degree \(3\), while the denominator has degree \(2\). Rewrite the numerator to separate a polynomial term:
Therefore, since \(x^2+1\ne 0\) for real \(x\),
Integrate the two terms. For the second term, \(x^2+1\) has derivative \(2x\), so a factor of \(\frac12\) is required:
Check by differentiating. Since \(x^2+1>0\), the logarithm is defined for every real \(x\):
The rewrite made the integral a sum of two standard forms. Trying to treat the original fraction as a single pattern would hide that simple structure.
Worked Example: Use the Composite Structure After Simplifying
Worked Example: A Trigonometric Product Over a Composite Denominator
Evaluate \(\int \frac{\sin x\cos x}{2+\sin^2 x}\,dx\).
The denominator is a function of \(\sin^2 x\), and the numerator contains \(\sin x\cos x\). The derivative of \(2+\sin^2 x\) is \(2\sin x\cos x\), so make the substitution
Thus \(\sin x\cos x\,dx=\frac12\,du\), and the integral becomes
For real \(x\), \(2+\sin^2x\ge 2\), so the absolute value can be removed when returning to \(x\):
Check by differentiating:
Here the product and denominator are not separate obstacles. Together, they reveal a composite function and its derivative.
Worked Example: Split the Numerator to Match Two Forms
Worked Example: A Shifted Quadratic Denominator
Evaluate \(\int \frac{2x+3}{(x+1)^2+4}\,dx\).
The denominator contains \(x+1\), and the numerator can be split into a multiple of \(x+1\) plus a constant:
Consequently,
The first term has a numerator equal to the derivative of the quadratic denominator. The second has the form of an arctangent antiderivative. Integrate each:
For the logarithm term, its argument is positive for every real \(x\). To check the result, differentiate each term:
Adding these derivatives gives
which is the original integrand.
Why Layering Works
The method rests on two basic facts. First, an algebraic identity allows an integrand to be replaced by an equal expression on its domain. Second, integration is linear: if \(f\) and \(g\) have antiderivatives on an interval, then a sum or difference can be integrated term by term. Substitution works when the differential in the integral matches the derivative of the chosen inner function.
This argument explains why the order matters. A rewrite is useful when it makes the chain-rule factor visible, separates terms that match known forms, or removes unnecessary algebra. A rewrite that does none of those things may add work without helping.
A Common Mistake: Losing a Constant Factor
Consider the second term in the first example. A common incorrect step is
Differentiating the proposed answer exposes the error:
which is twice the required integrand. Since \(d(x^2+1)=2x\,dx\), the correct result is
When a substitution or reverse chain-rule pattern appears, check the exact derivative factor, not just the shape of the inner function.
Choosing the Next Layer
After each rewrite, inspect the new integrand again. A first step may reveal that no second method is needed, as in the trigonometric example. In other cases, the rewrite produces a sum whose terms call for different standard forms, as in the shifted quadratic example. If a product remains and no direct substitution or identity resolves it, the product may be a signal to consider integration by parts. The appropriate method depends on the expression that remains, not only on how the original integrand first looked.
Also keep track of the domain. An algebraic cancellation is valid only where the cancelled factor is nonzero, and a logarithm requires a nonzero argument (or a positive argument when written without absolute values). An antiderivative must be considered on intervals where the original integrand is defined.
Check Your Understanding
For each question, identify the useful first step and explain why it helps.
- For \(\int \frac{x^5}{x^2+1}\,dx\), what algebraic operation could expose simpler terms?
- For \(\int \frac{3x^2}{1+x^3}\,dx\), which inner function would you test for substitution?
- In \(\int \frac{2x+1}{(x+1)^2+9}\,dx\), how might you split the numerator to reveal separate forms?
- Why must you check the domain when cancelling a factor in a rational integrand?
- How can differentiating a proposed antiderivative detect a missing constant factor?
