Introduction
Mixed integrands often look more complicated than the calculations they require. A product may hide a simple inner function and its derivative; a rational expression may become straightforward after division; a trigonometric power may become a polynomial after a useful substitution. The first task is therefore not to start integrating immediately. It is to read the structure of the integrand.
This tutorial develops a practical strategy for choosing among familiar methods. The aim is not to apply every method to every expression, but to make a small, justified rewrite, carry out the resulting integral, and then differentiate the proposed answer to check it.
A Short Decision Process
Before calculating, inspect the expression in a consistent order. First check whether ordinary algebra can simplify it. Then look for a function paired with its derivative, or for a standard pattern such as a polynomial, a trigonometric power, or a radical with a suitable substitution. If a proposed method does not simplify the integrand, reconsider it before proceeding.
Expand, factor, cancel, or divide only when the rewrite makes the integration more direct. Keep track of any restrictions introduced by denominators.
Check for a derivative paired with a composite expression, a trigonometric parity pattern, or a power that becomes simpler after substitution.
Compute the antiderivative, include the constant of integration, and differentiate the result to recover the original integrand.
These checks do not guarantee that every integral has a simple antiderivative. They do help distinguish a promising method from a rewrite that merely makes the expression longer.
Why Derivative Pairing Works
A common pattern is an outer function applied to an inner function, multiplied by the derivative of that inner function. The substitution method is justified by the chain rule: differentiating a composite function produces the derivative of the outer function multiplied by the derivative of the inner function.
For the conclusion, differentiate the proposed antiderivative \(F(g(x))+C\). The chain rule gives \(F'(g(x))g'(x)\), and the condition \(F'=f\) makes this \(f(g(x))g'(x)\), exactly the original integrand. A constant multiple of \(g'(x)\) can also be handled by factoring that constant out.
Worked Examples
Worked Example: Integrating a Polynomial Term by Term
Find an antiderivative of \(3x^2-4x+7\). The integrand is already a sum of powers of \(x\), so no substitution or algebraic rewrite is needed. Integrate each term using the power rule for antiderivatives:
Check by differentiation:
The derivative agrees with the integrand, so the antiderivative is verified.
Worked Example: Dividing Before Integrating a Rational Function
Find an antiderivative of \(\frac{x^2+3x+1}{x+1}\), on an interval where \(x\ne -1\). The numerator has degree two and the denominator has degree one, so polynomial division is a useful first step. Since \((x+1)(x+2)=x^2+3x+2\), the remainder is \(-1\):
Now integrate each term. The logarithm is written with an absolute value because the denominator can be positive or negative on the allowed intervals:
Check the result on either interval \(x<-1\) or \(x>-1\):
The division reduced the rational expression to a polynomial and one standard logarithmic form. There is no need to introduce a more involved decomposition when this simpler rewrite already resolves the integral.
Worked Example: Using a Radical’s Inner Derivative
Find an antiderivative of \(x\sqrt{x^2+5}\). The quantity under the radical has derivative \(2x\), which is a constant multiple of the factor \(x\). Set \(u=x^2+5\), so \(du=2x\,dx\) and \(x\,dx=\frac12\,du\). Then
Differentiate to check, applying the chain rule:
The inner derivative supplies exactly the factor needed, apart from a constant that is accounted for in the calculation.
Worked Example: Choosing a Substitution for Trigonometric Powers
Find an antiderivative of \(\sin^3 x\cos^2 x\). The sine power is odd. Separate one factor of \(\sin x\), then use \(\sin^2 x=1-\cos^2 x\):
Set \(u=\cos x\), so \(du=-\sin x\,dx\). The remaining expression is a polynomial in \(u\):
Check by differentiating both terms:
The substitution is useful because the odd sine factor supplies \(du\), while the remaining sine power can be rewritten using the identity.
Worked Example: Matching a Composite Power to Its Derivative
Find an antiderivative of \(4x(2x^2+1)^6\). The inner function \(2x^2+1\) has derivative \(4x\), which appears as the other factor. Set \(u=2x^2+1\), so \(du=4x\,dx\):
Differentiate to verify:
This is a direct reverse-chain-rule pattern: the inner derivative is present without any additional factor.
When a Promising Pattern Is Not Enough
Recognizing a composite expression is only the beginning. The derivative of the inner function must also be present, up to a constant multiple, for the direct substitution pattern above to apply as written. If it is missing, do not silently treat it as though it were there.
For example, \(e^{x^2+2x}\) has inner function \(x^2+2x\), whose derivative is \(2x+2\). The integrand \((x+1)e^{x^2+2x}\) contains half that derivative. Therefore the constant factor must be included:
The check confirms the factor:
Making the Strategy Efficient
The examples illustrate several distinct cues, but they share a useful habit: choose a method because of a visible structural match, not simply because the method is familiar. Polynomial division was effective for the rational example because it produced simpler terms. Substitution was effective in the radical and composite examples because the inner derivative was present. In the trigonometric example, parity suggested a factor to save and an identity to use.
Before committing to a longer calculation, ask whether the proposed rewrite reduces the difficulty. If a substitution creates more complicated powers or introduces factors that were not present, it may not be the right choice. If an algebraic expansion increases the number of terms without simplifying their antiderivatives, leave the expression factored. A useful strategy should make the next step clearer.
Verification is part of this process, not an optional final flourish. If differentiating the proposed answer fails to recover the integrand, the discrepancy often points directly to a missing constant, a sign error, or an incorrect rewrite. In a definite integral, the setup and the bounds also require care; the examples here concern antiderivatives, so no endpoint evaluation is needed.
Summary
A mixed integrand calls for inspection before calculation. Integrate a polynomial directly when it is already simple; consider division when a rational function can be reduced; use trigonometric identities when parity exposes a derivative pairing; and test radical or composite expressions for an inner derivative. These are choices guided by structure rather than a fixed formula.
Whatever the chosen method, finish by differentiating the candidate antiderivative. Agreement with the original integrand verifies the calculation on its domain and reinforces the reason the strategy worked.
Check Your Understanding
For each question, identify the useful structure before carrying out any calculation.
- What rewrite makes \(\frac{x^2+2x+3}{x+1}\) simpler to integrate, and what restriction applies to its domain?
- In \(\int 6x(x^2+4)^3\,dx\), which inner function suggests a substitution, and what is its derivative?
- For \(\sin^5 x\cos^2 x\), which factor would you separate before using a trigonometric identity? Why?
- What factor is needed in an antiderivative of \(x e^{x^2}\), and how can differentiation confirm it?
- Why is differentiating a proposed antiderivative a useful way to locate a missing-factor error?
