Common Mistakes and Notes

Most wrong answers in integration are not wrong because the student could not find an antiderivative. They are wrong because of a small thing attached to a correct one: a missing constant, a missing absolute value, a missing factor of $\tfrac{1}{2}$, a formula used outside the domain where it holds. The integrand was handled; the bookkeeping was not.

This page collects those failures, groups them by where they come from, and explains the reason behind each one. The point is not to memorize a list of prohibitions but to understand that every entry below is the shadow of a correct idea: the indefinite integral is a family, a logarithm needs a positive argument, a substitution changes the differential, and a formula has a domain. Once those four facts are solid, most of the errors stop happening on their own.

Definition

For a function $f$ continuous on an interval $I$, an antiderivative of $f$ is a function $F$ with $F'(x) = f(x)$ on $I$. The indefinite integral of $f$ is the family of all its antiderivatives:

$$\int f(x)\,dx = F(x) + C,$$

where $C$ is an arbitrary constant. Two facts follow immediately and account for most of what follows.

First, the indefinite integral is a set of functions, not a single function, so it must carry the constant $C$. Second, a definite integral $\int_a^b f(x)\,dx$ is a number, produced by evaluating an antiderivative at the two endpoints; the constants cancel, so a definite integral has no $C$.

Errors Involving the Constant of Integration

Forgetting the constant. Writing $\int x\,dx = \frac{x^2}{2}$ is wrong; the correct statement is $\int x\,dx = \frac{x^2}{2} + C$. The reason is definitional: $\frac{x^2}{2}$ is one antiderivative, while the indefinite integral denotes all of them, and $\frac{x^2}{2} + 7$ is an antiderivative too.

Adding a constant to a definite integral. Writing $\int_0^1 x\,dx = \frac{x^2}{2} + C$ is wrong; the correct value is $\int_0^1 x\,dx = \frac{1}{2}$. A definite integral is a single number, and an answer containing a free parameter is not a number at all. The same reasoning shows why the constant disappears legitimately when you evaluate an antiderivative at both endpoints.

How to avoid them. Decide first whether you are computing an indefinite or a definite integral, and let the answer’s type follow. Then differentiate the result: a constant vanishes under differentiation, so a missing $C$ is invisible to that check and must be caught by the habit of writing it down.

Errors Involving the Basic Formulas

Dropping the absolute value in a logarithm. Writing $\int \frac{1}{x}\,dx = \ln x + C$ is wrong; the correct statement is $\int \frac{1}{x}\,dx = \ln\lvert x\rvert + C$. The reason is the domain of the logarithm: $\ln x$ exists only for $x > 0$, so $\ln x$ is not an antiderivative of $\frac{1}{x}$ on the negative half-line, while $\ln\lvert x\rvert$ is defined for every $x \neq 0$ and has derivative $\frac{1}{x}$ on both halves.

Applying the power rule where it does not reach. Writing $\int x^{-1}\,dx = \frac{x^0}{0} + C$ is wrong; the correct statement is $\int x^{-1}\,dx = \ln\lvert x\rvert + C$. The power rule $\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$ requires $n \neq -1$, because at $n = -1$ the denominator $n+1$ vanishes.

Losing the chain-rule factor. Writing $\int e^{2x}\,dx = e^{2x} + C$ is wrong; the correct statement is $\int e^{2x}\,dx = \frac{1}{2}e^{2x} + C$. The factor $\tfrac{1}{2}$ comes from the chain rule: the derivative of $e^{2x}$ is $2e^{2x}$, so $\frac{1}{2}e^{2x}$ is the antiderivative of $e^{2x}$.

How to avoid them. Check the exponent and the coefficient before substituting into a formula, and confirm the result by differentiating it. For the logarithmic case the check is special and quick: differentiate $\ln\lvert x\rvert$ and you get $\frac{1}{x}$ on both halves of the line.

Errors Involving the Interval and the Domain

Integrating across a discontinuity. Writing $\int_{-1}^{1} \frac{1}{x}\,dx = \left[\ln\lvert x\rvert\right]_{-1}^1 = 0$ is wrong, and it is wrong in an instructive way: the arithmetic is correct and the integral still does not exist. The integrand has a vertical asymptote at $x = 0$, so the definite integral on $[-1,1]$ is not defined by the fundamental theorem at all. As an improper integral it diverges, since the left and right pieces both diverge.

Ignoring the domain of a formula. Writing $\int \frac{1}{\sqrt{1-x^2}}\,dx = \arcsin x + C$ without qualification is wrong; the formula holds only for $\lvert x\rvert < 1$, because outside that range the integrand is not real-valued. Similarly, $\int \frac{1}{\lvert x\rvert\sqrt{x^2-1}}\,dx = \arccos x + C$ holds for $\lvert x\rvert > 1$.

How to avoid them. Before writing the integral, list the points where the integrand is undefined or discontinuous, and decide whether the integral is proper or improper. Whenever a formula is applied, state the interval on which it is being used. It is worth remembering that the fundamental theorem of calculus requires continuity on the closed interval of integration; that requirement is exactly what the first error violates.

Errors Involving Technique

Applying a formula that does not match. Writing $\int \frac{1}{x^2+1}\,dx = \frac{1}{2x} + C$ is wrong; the correct statement is $\int \frac{1}{x^2+1}\,dx = \arctan x + C$. The integrand was not recognized as the derivative of an inverse tangent, and a formula was forced onto it. The result is easily caught by differentiation: the derivative of $\frac{1}{2x}$ is $-\frac{1}{2x^2}$, which has nothing to do with the integrand.

Losing the differential in a substitution. Writing $\int x e^{x^2}\,dx = \int e^u\,du = e^u + C$ is wrong; the correct statement is $\int x e^{x^2}\,dx = \frac{1}{2}e^{x^2} + C$. The substitution $u = x^2$ gives $du = 2x\,dx$, so $x\,dx = \frac{1}{2}du$, and that factor $\tfrac{1}{2}$ has to travel with the differential into the new integral.

Forgetting to back-substitute. Leaving the answer as $\int x e^{x^2}\,dx = \frac{1}{2}e^u + C$ is wrong; the answer must be expressed in the original variable, so it is $\frac{1}{2}e^{x^2} + C$. An answer in the auxiliary variable is not an answer to the question that was asked.

How to avoid them. Read the integrand before reaching for a method: its algebraic shape decides which technique is even possible. In a substitution, rewrite the entire integral including the differential, and if the integral is definite, transform the limits as well. Finally, always back-substitute before writing the final line.

Arithmetic Errors

Coefficient errors. Writing $\int (2x^2 + x)\,dx = x^3 + \frac{x^2}{2} + C$ is wrong; the correct statement is $\int (2x^2 + x)\,dx = \frac{2x^3}{3} + \frac{x^2}{2} + C$. The coefficient $2$ was dropped when the term $\int 2x^2\,dx$ was integrated by the power rule, which divides by $n+1 = 3$.

Sign errors. Writing $\int -x\,dx = \frac{x^2}{2} + C$ is wrong; the correct statement is $\int -x\,dx = -\frac{x^2}{2} + C$. A negative sign in front of the integrand must survive into the answer.

Exponent errors. Writing $\int x^2\,dx = x^3 + C$ is wrong; the correct statement is $\int x^2\,dx = \frac{x^3}{3} + C$. The power rule raises the exponent by one and divides by the new exponent.

How to avoid them. Write each algebraic step on its own line with the coefficients still visible, and then differentiate the finished expression. That single check catches coefficient errors, sign errors and exponent errors at once, because the derivative of $\frac{2x^3}{3} + \frac{x^2}{2}$ is $2x^2 + x$ while the derivative of the faulty answer is $3x^2 + x$.

Tips for Memorising the Table of Integrals

1. Group the formulas by the type of function

Power functions: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ ($n \neq -1$)

Exponential functions: $\int e^x dx = e^x + C$

Logarithmic functions: $\int \frac{1}{x} dx = \ln\lvert x\rvert + C$

Trigonometric functions: $\int \sin x dx = -\cos x + C$, $\int \cos x dx = \sin x + C$

2. How to remember them

The relation between differentiation and integration:

• Integration is the inverse operation of differentiation
• The result of an integral can be checked by differentiating it

The geometric meaning:

• An integral represents an area or a volume
• An indefinite integral represents a family of curves

The physical meaning:

• An integral represents a cumulative quantity
• For instance, the displacement is the integral of the velocity

A Checklist That Catches Almost Everything

1. Is this indefinite or definite? If indefinite, is there a $+C$? If definite, is there no free constant?
2. Does the domain allow this? Are all absolute values present, and is the formula valid on the interval in question?
3. Did the substitution carry its factor and get back-substituted?
4. Does differentiating the answer reproduce the integrand?

The fourth step is the powerful one, because it verifies the whole chain at once and, unlike a visual check, it cannot be fooled by an answer that merely looks plausible.

Exercises

Exercise 1

Find and correct the error in the following computation: $\int \frac{1}{x}\,dx = \ln x + C$.

Exercise 2

Find and correct the error in the following computation: $\int x^{-1}\,dx = \frac{x^0}{0} + C$.

Exercise 3

Find and correct the error in the following computation: $\int e^{3x}\,dx = e^{3x} + C$.

Exercise 4

Determine whether the computation $\int (x^2 + 2x)\,dx = \frac{x^3}{3} + x^2 + C$ is correct.

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Summary

The errors collected on this page are not random. Each of them comes from one of four ideas being applied slightly wrong: the indefinite integral is a family and therefore carries $C$; the logarithm needs a positive argument, hence $\ln\lvert x\rvert$ rather than $\ln x$; a substitution changes the differential, so a factor must travel with it and the original variable must come back at the end; and every formula has a domain on which it is valid. Where the arithmetic itself fails, the same discipline applies, since a missing coefficient, a lost sign and a wrong exponent are all detected by differentiating the result.

The habits that prevent these errors are few and worth stating plainly. Decide whether the integral is definite or indefinite before writing it. State the domain when you use a formula. Carry the differential through a substitution and back-substitute afterwards. And then differentiate the answer: it is the only single check that catches every algebraic error on this page at once, because it compares the finished result with the original integrand rather than with an intermediate step.

Symbols Used in This Article