Four derivative matches and one deliberately free constantLive current inputs
Math & Statistics
Integral Verification Calculator
Verify whether a proposed quartic antiderivative differentiates to an entered cubic integrand, report coefficient residuals independently, and test endpoint accumulation despite an arbitrary constant.
Antiderivative verdict -
Maximum coefficient residual -
Cubic-term residual -
Quadratic-term residual -
Linear-term residual -
Constant-term residual -
Integration constant status -
Terms outside tolerance -
ANTIDERIVATIVE COEFFICIENT GATES
Four derivative matches and one deliberately free constant
Each coefficient residual receives its own tolerance rail; the proposed integration constant is shown outside the pass/fail system because differentiation removes it.
TERM-BY-TERM AUDIT
Differentiate each proposed term and compare like powers
The table aligns derivative coefficients by power of x, preventing a correct total at one x-value from hiding a wrong antiderivative.
| Power in f | Required coefficient | From proposed F' | Residual | Tolerance status |
|---|
VERIFICATION SETUP
Enter the proposal exactly as written, including zero terms
- Transcribe the cubic integrand coefficients by descending power.
- Transcribe the proposed quartic antiderivative the same way.
- Keep E even though it will not affect the verdict.
- Choose tolerance based on coefficient precision, not function values at one test point.
- Correct only the failed powers identified in the table.
WHY COEFFICIENTS
Matching at one point is not the same as matching as functions
Two different polynomials can have the same value at one x-coordinate. Coefficient comparison after differentiation verifies the full polynomial identity.
The constant E represents the entire family of antiderivatives. Treating one specific E as required would incorrectly reject valid solutions.
COEFFICIENT VERIFICATION
Differentiate the proposal instead of re-integrating the target
If F is an antiderivative of f, then F' must match every coefficient of f. The additive constant E is unconstrained because its derivative is zero.
Detailed calculation process and general formulas
f(x)=ax^3+bx^2+cx+dF(x)=Ax^4+Bx^3+Cx^2+Dx+EF'(x)=4Ax^3+3Bx^2+2Cx+Dr=[4A-a, 3B-b, 2C-c, D-d]pass iff max |r_i| <= tolSymbols, meanings, and units
- a,b,c,d
- required integrand coefficientspower-specific
- A,B,C,D
- proposed antiderivative coefficientspower-specific
- E
- arbitrary integration constantF-unit
- r_i
- coefficient residual after differentiationintegrand coefficient
- tol
- allowed absolute coefficient residualintegrand coefficient
CORRECTION GUIDE
Each failed gate points to one integration step
The derivative multiplier identifies the corresponding antiderivative divisor.
Cubic mismatch
-Correct A to a divided by four.
Quadratic mismatch
-Correct B to b divided by three.
Lower-power mismatch
-Inspect C=c/2 and D=d independently.
Decision takeaway: Fix the term named by the residual instead of adjusting the free constant.
AUDIT TRAPS
Mistakes this layout is designed to expose
- Forgetting to raise the power during integration
- Dividing by the old power rather than the new power
- Dropping the original constant integrand term
- Trying to use E to repair a derivative mismatch
- Checking only one numerical x-value
Applied decisions
Antiderivative reviews with different failure signatures
Single divisor error
Only the quartic coefficient is divided by three instead of four.
What the result clarifies: The cubic residual fails while every lower gate remains green.
Different integration constants
Two correct antiderivatives differ only in E.
What the result clarifies: Both pass because the free constant does not alter F'.
Worked default scenario
Current-input substitution and reconciliation
Method references
References for this calculator's specific method
Scope and limitations
This calculator verifies polynomial antiderivative coefficients within an absolute tolerance. It does not verify domain restrictions, piecewise definitions, or non-polynomial expressions.
Integral Verification Calculator | Antiderivative Coefficient Audit FAQ
Why is E never marked wrong?
The derivative of any constant is zero, so every E belongs to the same antiderivative family.
Can a proposal pass at one x but fail here?
Yes. Point agreement is weaker than polynomial coefficient identity.
Should tolerance be zero?
Use zero for exact entered coefficients or a small justified tolerance for rounded decimals.
What does a cubic residual correspond to?
It tests whether 4A equals the integrand's cubic coefficient a.