Why Check an Antiderivative?
An indefinite integral asks for a function whose derivative is a given integrand. The result can be easy to misremember: a chain-rule factor may be missing, a negative sign may be wrong, or a power may have been increased or decreased incorrectly. Differentiation provides a direct check. Instead of relying on how an answer looks, test whether its derivative is exactly the original integrand.
This check is useful both after completing an integral and while diagnosing a suspected error. If the derivative does not match, comparing the two expressions often shows precisely what needs to change.
The Derivative Test
Suppose an indefinite integral is written as \(\int f(x)\,dx\), and \(F(x)+C\) is proposed as its value. To check the function part, compute \(\frac{d}{dx}F(x)\). If the result simplifies to \(f(x)\), then \(F\) is an antiderivative of the integrand on the domain being considered. The constant \(C\) does not affect the check, because the derivative of a constant is zero.
The comparison must be exact, not merely similar. A factor of \(5\), a minus sign, or a different power means the candidate does not pass the test unless algebraic simplification shows the expressions are equal.
Apply the relevant derivative rules, including the chain rule for composite functions.
Rewrite it in a form that can be compared directly with the original integrand.
If the expressions agree on the interval, the proposed function is verified. If they differ, identify the factor, sign, or power responsible.
Make the needed change, then differentiate the corrected candidate to confirm the result.
Why the Chain Rule Exposes Missing Factors
Many integration errors involve a composite function. The chain rule explains why checking by differentiation is especially effective in these cases. If a candidate has the form \(G(g(x))\), then differentiating it produces the derivative of the outer function, evaluated at the inner function, multiplied by the derivative of that inner function.
For an antiderivative candidate to be correct, this entire product must equal the integrand. A missing numerical factor in the candidate remains visible after differentiation, while an omitted inner-function derivative usually appears as a missing factor of \(g'(x)\). This is the reverse-chain-rule idea behind many substitution integrals, now used as a verification method.
Worked Checks
Worked Example: Checking a Polynomial Antiderivative
Check whether \(F(x)=x^3-2x^2+2x\) is an antiderivative of \(f(x)=3x^2-4x+2\).
Differentiate each term using the power rule:
The derivative equals the original integrand \(3x^2-4x+2\). Therefore \(F\) passes the derivative test and is an antiderivative of \(f\). If the proposed indefinite integral includes \(+C\), differentiating that term gives zero, so the check is unchanged.
Check: Substituting the derivative into the comparison gives \(F'(x)=3x^2-4x+2=f(x)\), as required.
Worked Example: Finding a Missing Chain-Rule Factor
A proposed answer to \(\int e^{5x}\,dx\) is \(F(x)=e^{5x}\). Check it, then correct it if necessary.
Differentiate the proposed function. The inner function is \(5x\), whose derivative is \(5\):
The integrand is \(e^{5x}\), not \(5e^{5x}\). The proposed answer has an extra factor of \(5\) in its derivative, so it fails the test. To correct it, multiply the candidate by \(\frac{1}{5}\):
Now differentiate the corrected candidate:
Check: The corrected derivative is exactly the original integrand, so \(\frac{1}{5}e^{5x}\) is verified.
Worked Example: Detecting an Exponent Error
Check the proposed antiderivative \(F(x)=(x^2+1)^2\) for the integrand \(f(x)=6x(x^2+1)^2\), then find a correct candidate.
Differentiate the proposed function with the chain rule. The outer power is \(2\), and the derivative of the inner function \(x^2+1\) is \(2x\):
This is not \(6x(x^2+1)^2\): the coefficient is different, and the power of \(x^2+1\) is also different. Thus the proposed exponent is wrong for this integrand. Try instead \(G(x)=(x^2+1)^3\). Differentiating gives
Check: The final derivative matches \(f(x)\) exactly, so \(G(x)=(x^2+1)^3\) is a correct antiderivative.
Worked Example: Correcting a Sign Error
Check the proposed antiderivative \(F(x)=\frac{1}{3}\cos(3x)\) for the integrand \(f(x)=\sin(3x)\).
Differentiate. The derivative of cosine introduces a negative sign, and the inner function \(3x\) contributes a factor of \(3\):
The derivative has the opposite sign from the integrand, so the proposed function is not an antiderivative of \(\sin(3x)\). Change the sign of the candidate and check again:
Check: The corrected derivative equals the given integrand, including its positive sign. The corrected candidate is verified.
Reading the Difference
When a candidate fails, the form of its derivative often points directly to the error. Compare the derivative with the integrand term by term or factor by factor. For a composite function, look for an inner derivative that is absent or incorrectly scaled. For trigonometric functions, check the derivative's sign as well as its argument. For powers, verify that differentiating the proposed exponent produces the power and coefficient required by the integrand.
| What the derivative shows | Likely issue to inspect |
|---|---|
| The derivative has an extra numerical factor. | A reciprocal factor may be missing from the candidate. |
| The derivative differs by a negative sign. | The sign of the candidate may need to change. |
| The derivative has the wrong power of a composite expression. | Recheck the proposed exponent using the power rule and chain rule. |
| The derivative contains an unexpected inner-function factor. | Check whether the chain-rule factor was accounted for. |
A Common Mistake
A common mistake is to differentiate only the outside of a composite function. For example, someone might check \(e^{5x}\) by writing that the derivative of \(e^u\) is \(e^u\), and conclude that the derivative is \(e^{5x}\). That omits the derivative of the inner function \(u=5x\).
The correction is to apply the chain rule to the whole composite expression. A reliable check writes the inner derivative explicitly before simplifying. This prevents a missing factor from being hidden by a familiar-looking outer derivative.
Why Verification Matters
Differentiating a proposed antiderivative does more than confirm a final answer. It checks that the answer reverses the differentiation rules correctly, including the chain rule. It can also turn an uncertain memory into a concrete test: rather than deciding whether a formula looks plausible, you establish whether its derivative is the required function.
For AP Calculus work, make the comparison visible. Show the derivative, simplify it, and state whether it agrees with the integrand. If it does not, name the mismatch and verify the correction. A compact check is most useful when it includes all factors and signs rather than skipping the part most likely to contain the error.
Check Your Understanding
For each question, use differentiation to test the proposed function or explain what must be corrected.
- Is \(F(x)=2x^3-5x\) an antiderivative of \(6x^2-5\)? Show the derivative.
- A proposed antiderivative of \(e^{4x}\) is \(e^{4x}\). What does differentiation reveal, and what coefficient corrects the candidate?
- Check whether \(F(x)=\frac{1}{2}\sin(2x)\) is an antiderivative of \(\cos(2x)\).
- Differentiate \(F(x)=(x^2+3)^4\). What power and factors appear in its derivative?
- A candidate differentiates to \(-\sin(5x)\), but the integrand is \(\sin(5x)\). What change to the candidate would correct the sign?
