The fastest way to check many integration answers is to differentiate them. Integration constructs an antiderivative; differentiation tests whether that antiderivative returns the original integrand.
This reverse relationship is one of the strongest checking tools in Additional Mathematics because it tests the mathematical meaning of the answer rather than only its appearance.
1. The core reverse check
If you claim
∫f(x)dx=F(x)+C,
then differentiating F(x)+C should give exactly f(x).
If it does not, the integration is wrong or has been simplified incorrectly.
2. Worked polynomial check
Suppose
∫(12x³−6x+4)dx=3x⁴−3x²+4x+C.
Differentiate:
12x³−6x+4.
The original integrand is recovered exactly, so the antiderivative passes the reverse check.
3. The reverse check exposes missing divisors
A student writes ∫x⁴dx=x⁵+C. Differentiate the claimed answer: 5x⁴, not x⁴.
The check immediately reveals the missing division by 5. The correct result is x⁵/5+C.
4. Check signs in trigonometric integrals
Suppose ∫sin x dx is claimed to be cos x+C. Differentiating cos x gives −sin x, so the sign is wrong.
The correct antiderivative is −cos x+C.
5. Check reverse chain-rule patterns
Consider the proposed antiderivative
F(x)=(2x²+1)^5/10.
Differentiate:
F′(x)=(1/10)·5(2x²+1)^4·4x=2x(2x²+1)^4.
Therefore F is an antiderivative of 2x(2x²+1)^4, not 4x(2x²+1)^4. The reverse derivative exposes the scale mismatch.
6. Check +C separately
Differentiating C gives zero, so the reverse derivative cannot tell you whether you remembered to write +C. This part must be checked conceptually.
For an indefinite integral, ask: “Have I represented the full family of antiderivatives?” If not, add +C.
7. Conditions check the constant
Suppose dy/dx=2x+1 and the curve passes through (1,5).
Integration gives y=x²+x+C. The point gives 5=1+1+C, so C=3.
Two checks are available:
- Differentiate y=x²+x+3 to recover 2x+1.
- Substitute x=1 to recover y=5.
Passing both checks tests the derivative relationship and the selected constant.
8. Definite integrals need a different final check
For ∫_a^b f(x)dx, differentiating the final numerical answer is meaningless. Instead, check the antiderivative before substituting limits, then verify that the upper-minus-lower evaluation is correct.
A graph or sign check can then test whether the numerical result is plausible.
9. Worked definite-integral check
Evaluate ∫_0^2 (2x+3)dx.
Antiderivative F=x²+3x. Differentiate F to verify F′=2x+3.
Then evaluate F(2)−F(0)=4+6=10.
Graph check: the line has heights 3 and 7 over width 2, so trapezium area=(1/2)(3+7)(2)=10. Three independent representations agree.
10. Graph sign checks
If f(x)>0 throughout [a,b], then ∫_a^b f(x)dx should be positive. A negative answer is a warning that the limits may have been reversed or an algebraic sign error occurred.
If the graph crosses the axis, the sign of the net integral should reflect the balance of positive and negative regions.
11. Units check definite integrals
If f is a velocity in m/s and x is time in seconds, the integral should have units metres. If your answer is reported in m/s, you have retained the rate units instead of the accumulated-quantity units.
12. Domain checks still matter
For ∫1/x dx=ln|x|+C, the integrand is undefined at x=0. A definite integral crossing zero cannot be treated as an ordinary proper integral using one antiderivative substitution without addressing that discontinuity.
At school level, questions are normally designed with appropriate domains, but the principle remains important: an antiderivative formula does not erase a break in the original function.
13. Equivalent antiderivatives can look different
Two antiderivatives of the same function may differ only by a constant. For example, x²+5 and x²−9 both differentiate to 2x.
If two answers differ by a constant in an indefinite-integration setting, they may represent the same antiderivative family once +C is included.
14. Check by differentiating before heavy simplification
If an antiderivative is correct in a factored form, you do not need to expand it merely to make it look conventional. Differentiating the factored form may be easier and avoids introducing new algebra errors.
15. Error diagnosis by first unsafe line
When a final integral is wrong, locate the earliest step where the antiderivative stopped differentiating back to the integrand.
- Wrong power increase → power-rule integration error.
- Missing division → coefficient error.
- Wrong trig sign → antiderivative recall error.
- Missing inner factor → reverse-chain mismatch.
- Correct antiderivative, wrong limits → evaluation error.
- Correct integral, wrong geometric interpretation → signed-area error.
16. A 30-second integration checklist
- Differentiate the antiderivative mentally or on paper.
- Does it reproduce every term and coefficient?
- For an indefinite integral, is +C present?
- For a definite integral, did I use upper minus lower?
- Does the sign match the graph or context?
- Do the units represent accumulation rather than rate?
17. Independent practice
- Check whether x⁴/4+C is an antiderivative of x³.
- Check whether −sin x+C is an antiderivative of cos x.
- Find and check ∫(8x³+2)dx.
- Evaluate ∫_1^2 3x²dx and check the antiderivative first.
- If dy/dx=4x and y(2)=11, find y and perform both checks.
- Explain why a negative definite integral can still correspond to a positive geometric area.
18. Answers
- Yes. d/dx(x⁴/4)=x³.
- No. d/dx(−sin x)=−cos x. An antiderivative of cos x is sin x+C.
- 2x⁴+2x+C; differentiation returns 8x³+2.
- Antiderivative x³; [x³]_1^2=8−1=7.
- y=2x²+C. 11=8+C, so C=3 and y=2x²+3. Differentiate to get 4x; substitute x=2 to get 11.
- A below-axis region contributes negatively to signed area; geometric area uses magnitude.
19. What mastery looks like
A strong integration workflow produces an antiderivative and immediately asks whether differentiation returns the original function. It then checks constants, limits, signs, units and domain before using the result in area, motion or reconstruction problems.
Continue with What Integration Means, Definite Integration, or return to the Additional Mathematics Hub.