The most common antiderivative mistakes are: forgetting \(+C\), using the power rule on \(\frac1x\), forgetting to divide by the inside’s slope (as in \(\int\cos 4x\,dx\)), getting the sign of \(\int \sin x\,dx\) wrong, splitting a product into two integrals, writing \(\ln\) for \(\int\frac{1}{1+x^2}\,dx\), and inventing a “reverse chain rule”. One habit catches almost all of them: differentiate your answer.
This page works like a “spot the error” worksheet. Each section shows a wrong answer that looks believable, explains exactly what went wrong, gives the correct work, and names the quick check that would have caught it. Try to find each mistake yourself before you read the explanation.
Mistake 1: Forgetting + C
Wrong: \(\int 3x^2\,dx = x^3\)
What went wrong: \(x^3 + 1\), \(x^3 – 4\) and every other \(x^3 + C\) also have derivative \(3x^2\). Leaving out \(C\) names only one of them.
Correct: \(\int 3x^2\,dx = x^3 + C\).
The check: Ask “is this an indefinite integral?” If there are no limits, there must be a \(+C\). Read why +C matters, especially in initial value problems, where forgetting it gives a completely wrong function.
Mistake 2: Using the Power Rule on 1/x
Wrong: \(\int x^{-1}\,dx = \frac{x^0}{0}\) or “\(= x^0 = 1\)”
What went wrong: The power rule divides by \(n + 1\), which is 0 when \(n = -1\).
Correct: \(\int \frac1x\,dx = \ln|x| + C\).
The check: Before using the power rule, look at the exponent. If it is \(-1\), switch to the log rule. More in the antiderivative of 1/x.
Mistake 3: Forgetting to Divide by the Inside’s Slope
Wrong: \(\int \cos 4x\,dx = \sin 4x + C\)
What went wrong: Differentiate the wrong answer: \(\frac{d}{dx}\sin 4x = 4\cos 4x\). That is 4 times too big.
Correct: \(\int \cos 4x\,dx = \frac14\sin 4x + C\). The same goes for \(\int e^{5x}\,dx = \frac15e^{5x} + C\) and \(\int (3x – 1)^4\,dx = \frac{(3x-1)^5}{15} + C\).
The check: If the inside is \(ax + b\), the answer must have a \(\frac1a\) in front.
Mistake 4: The Wrong Sign on sin x
Wrong: \(\int \sin x\,dx = \cos x + C\)
What went wrong: \(\frac{d}{dx}\cos x = -\sin x\), not \(\sin x\).
Correct: \(\int \sin x\,dx = -\cos x + C\), while \(\int \cos x\,dx = \sin x + C\) has no minus sign.
The check: Use the cycle \(\sin \to \cos \to -\sin \to -\cos \to \sin\): differentiate by stepping right, integrate by stepping left. See the antiderivative of sin x.
Mistake 5: Splitting a Product Into Two Integrals
Wrong: \(\int x\cos x\,dx = \int x\,dx\cdot\int\cos x\,dx = \frac{x^2}{2}\sin x + C\)
What went wrong: There is no product rule for integrals. Differentiating \(\frac{x^2}{2}\sin x\) gives \(x\sin x + \frac{x^2}{2}\cos x\), nothing like \(x\cos x\).
Correct: Use integration by parts with \(u = x\): \(\int x\cos x\,dx = x\sin x + \cos x + C\).
The check: Sums can be split; products cannot. If you see two functions multiplied, reach for substitution or parts.
Mistake 6: Writing ln for Everything With a Fraction
Wrong: \(\int \frac{1}{1 + x^2}\,dx = \ln\left(1 + x^2\right) + C\)
What went wrong: The log rule only works when the top is the derivative of the bottom. Here the bottom’s derivative is \(2x\), and the top is just 1.
Correct: \(\int \frac{1}{1 + x^2}\,dx = \tan^{-1}x + C\). (By contrast, \(\int \frac{2x}{1+x^2}\,dx = \ln(1 + x^2) + C\) is right.)
The check: Before writing \(\ln|\text{bottom}|\), differentiate the bottom and compare it with the top.
Mistake 7: Inventing a “Reverse Chain Rule”
Wrong: \(\int e^{x^2}\,dx = \frac{e^{x^2}}{2x} + C\)
What went wrong: You cannot divide by a variable to undo the chain rule. Differentiating the wrong answer needs the quotient rule and does not give back \(e^{x^2}\). In fact \(e^{x^2}\) has no elementary antiderivative at all.
Correct: Substitution only works when the inside’s derivative is already a factor: \(\int 2x\,e^{x^2}\,dx = e^{x^2} + C\). For \(\int e^{x^2}\,dx\), use numbers instead of a formula. See what calculators can and cannot integrate.
The check: You may divide by a constant to fix a substitution. Never divide by \(x\).
One Habit That Catches Almost Every Mistake
Differentiate your answer. It takes about 20 seconds. If you get back the original function, you are right. If you do not, the leftover tells you which mistake you made: an extra factor of 4 means Mistake 3, a wrong sign means Mistake 4, and so on. The antiderivative calculator shows this check on every problem.
A few more slips that cost points: changing the variable in a definite integral without changing the limits, mixing up degrees and radians, and dropping a minus sign in the second round of integration by parts.
Test Yourself: Spot the Mistake
1. \(\int 6x^5\,dx = 30x^4 + C\)
That is the derivative, not the antiderivative. Correct: \(x^6 + C\).
2. \(\int e^{-2x}\,dx = e^{-2x} + C\)
Forgot to divide by \(-2\). Correct: \(-\frac12e^{-2x} + C\).
3. \(\int \frac{3}{x}\,dx = 3x^{0} + C\)
Power rule on \(\frac1x\). Correct: \(3\ln|x| + C\).
4. \(\int \sec^2 x\,dx = \sec x\tan x + C\)
Confused the derivative table. Correct: \(\tan x + C\).
Calculators for This Topic
Practice what you just read. Each tool shows full steps and checks its own answer.
- ∫Antiderivative Calculator – Find any antiderivative with full steps and a check.
- d/dxDerivative Calculator – Product, quotient and chain rule steps for any function.
- =Simplify Calculator – Simplified, expanded and factored forms.
Frequently Asked Questions
What is the most common antiderivative mistake?
Forgetting \(+C\) on indefinite integrals. The next most common is forgetting to divide by the coefficient of \(x\) inside functions like \(\cos 4x\) or \(e^{5x}\).
How can I avoid mistakes when integrating?
Simplify first, choose the method deliberately, write each step, and always differentiate your final answer to check it.
Is there a product rule for integration?
No. The integral of a product is not the product of the integrals. Use integration by parts or a substitution instead.
Why is my answer marked wrong when it differentiates correctly?
It might be missing \(+C\), or your teacher may want a simplified form. If your derivative matches the integrand, the math is right; only the form needs adjusting.
