To check an antiderivative, differentiate it. If the result equals the original expression that you integrated, your answer is correct.
This works for every lesson in integration methods, and it takes less than a minute. It is the most reliable habit you can build for an exam.
Why does checking work?
Integration and differentiation undo each other. If F(x) is an antiderivative of f(x), then F′(x) = f(x) by definition.
The constant c does not matter here, because the derivative of any constant is zero. So the check confirms everything except c, and c is confirmed separately with the given point.
How do you check, step by step?
- Write your candidate antiderivative F(x).
- Differentiate it term by term or with the chain rule.
- Compare with the original expression.
- If it differs, look at how: a constant multiple means a missed division, and a wrong sign means a power slip.
- For a curve through a point, substitute the point into the final equation as well.
Worked example
A student claims that ∫(2x + 1)³ dx = (2x + 1)⁴/4 + c. Test this, and correct it if needed.
Step 1, differentiate the candidate: d/dx of (2x + 1)⁴/4 = 4(2x + 1)³ × 2 / 4 = 2(2x + 1)³.
Step 2, compare: the result is 2(2x + 1)³, which is twice the original (2x + 1)³.
Step 3, find the cause: the chain rule gave an extra factor of 2, so the student forgot to divide by the inner coefficient.
Step 4, correct: use (2x + 1)⁴/8 + c.
Step 5, re-check: 4(2x + 1)³ × 2 / 8 = (2x + 1)³. It matches.
(2x + 1)⁴/8 + c
The size of the error tells you what went wrong. Twice as large means a missing division by 2.
The mistake to watch for
A common slip is to check by integrating again, or by trusting that the answer “looks right”.
Mistaken approach: “I integrated carefully, so it must be correct.”
Carefulness is not a check. The same misunderstanding will repeat if you work the same way a second time.
The correction is to use a different operation. Differentiating is a separate calculation from integrating, so an error in one is unlikely to hide in the other. Make the derivative the last line of your working.
Check yourself
Try these on paper, then open each answer.
1. Is ∫(x² + 2x) dx = x³/3 + x² + c correct?
Show answer
Differentiate: x² + 2x. This matches, so it is correct.
2. A student writes ∫6x(x² + 1)² dx = (x² + 1)³/3 + c. Check it and correct it.
Show answer
Differentiate: 3(x² + 1)² × 2x / 3 = 2x(x² + 1)². The original is 6x(x² + 1)², so the answer is three times too small. The correct integral is (x² + 1)³ + c, which differentiates to 3(x² + 1)² × 2x = 6x(x² + 1)².
3. A student writes ∫cos(2x) dx = 2 sin(2x) + c. Check it and correct it.
Show answer
Differentiate: 2 cos(2x) × 2 = 4 cos(2x), which is four times the original. The factor should have been divided, not multiplied. The correct answer is ½ sin(2x) + c.
Where this leads next
With checking in place, test yourself on the integration methods practice set. The calculus shape and rate explorer lets you compare a function and its derivative, and the non-calculator working trainer supports the arithmetic.
If you want someone to read your working and point out which habit is costing marks, our teachers do that in online one-to-one Additional Mathematics tuition.