Mistakes · Calculus
6 Common Integration Mistakes
Five of the six are caught the same way: differentiate your answer and compare it to the original expression.
1. Forgetting the +C
The most distinctive of the common integration mistakes, because differentiation has no equivalent. x³ and x³ + 100 have the same derivative, 3x², so integrating 3x² cannot recover which one you started with — the constant of integration is that missing piece, honestly written down rather than assumed.
A definite integral is the one place the +C is correctly absent, because it cancels when you subtract F(a) from F(b). Dropping it from an indefinite integral is not the same shortcut; there the constant never had anywhere to cancel to.
2. Dividing by the old exponent instead of the new one
The power rule raises the exponent by one and divides by the new exponent — the one on the answer, not the one you started with.
| Integral | Correct | The usual wrong version |
|---|---|---|
| ∫ x² dx | x³/3 | x³/2 |
| ∫ x⁵ dx | x⁶/6 | x⁶/5 |
| ∫ x dx | x²/2 | x²/1 |
It survives under time pressure because the exponent you last looked at was the old one. The fix is mechanical: write the new exponent down first, then divide by that number. The same swap trips people up on a negative exponent too — x⁻² integrates by raising the exponent to ⁻¹ and dividing by ⁻¹, not by the ⁻² you started from.
3. Using the power rule on 1/x
1/x is x⁻¹. Applying the power rule anyway raises the exponent to 0 and asks you to divide by 0, which is undefined — the rule genuinely does not apply here, and it is not an oversight in the rule but the reason a separate logarithm formula exists.
4. Treating a non-linear inside like a linear one
∫ sin(2x) dx works by dividing by the coefficient 2, because 2x is linear. ∫ sin(x²) dx does not follow the same shortcut — there is no constant coefficient of x to divide by, and differentiating cos(x²) would bring down 2x by the chain rule, not a constant, so 'divide by the coefficient' has nothing to act on.
The test before reaching for the linear-substitution shortcut: is what's inside the function of the form ax + b? If x appears with a power, inside another function, or multiplied by something that also contains x, the shortcut does not apply.
5. Splitting a product into two separate integrals
∫ uv dx is not ∫u dx · ∫v dx. Integration does not distribute over multiplication the way it distributes over addition.
One example settles it. ∫ x dx = x²/2, so treating x·x as a product of integrals would give (x²/2)² — a fourth-degree expression, when the correct answer to ∫ x² dx is x³/3, degree three. The shapes do not even match.
6. Forgetting to divide by the coefficient in linear substitution
∫ sin(2x) dx = −cos(2x) is right except for one missing factor: divide by 2. That factor exists to cancel the 2 the chain rule would reintroduce when the answer is differentiated back.
- d/dx [−cos(2x)] = 2sin(2x)differentiating the wrong answer
- 2sin(2x) ≠ sin(2x)it does not match the original integrand — off by the missing factor
- d/dx [−cos(2x)/2] = sin(2x)the corrected version matches exactly
This is the check worth running on every answer, not only this one: differentiate what you found, and it must reproduce the original expression with nothing left over.
The habit that catches five of the six
Always check your answer by differentiating it. It costs one extra line of work and it is not optional the way it might feel — differentiation is mechanical and unambiguous, so if the result does not simplify back to the original integrand, something above it is wrong, no exceptions.
The one mistake this habit cannot catch is the first: forgetting the +C entirely, since a missing constant differentiates to the same zero either way. That is the single reason it is listed on its own and worth a deliberate second look rather than a differentiation check.
Questions about common integration mistakes
Why do I need to write +C on an indefinite integral?
Because differentiation destroys constants — x³ and x³ + 100 share the same derivative. The +C represents whichever constant integration cannot recover.
What is the integral of x²?
x³/3. Divide by the new exponent, 3, not the old one, 2 — that swap is the most common power-rule mistake in integration.
Why doesn't the power rule work for 1/x?
The power rule would divide by zero, since raising the exponent −1 by one gives 0. A separate formula, ln|x| + C, covers this one case.
How do I check an integral without a calculator?
Differentiate your answer. If it reproduces the original expression exactly, nothing was dropped or mishandled — this check catches essentially every mistake above.