Calculus I · Unit 3A · lesson

Indefinite Integrals and the Constant of Integration

Concept

Learning objectives

Use indefinite-integral notation correctly, distinguish an integrand from an antiderivative, and carry the constant of integration through calculations.

Indefinite Integrals and the Constant of Integration

Explanation

Reading indefinite-integral notation correctly

An indefinite integral is not a number and it is not an unfinished definite integral. It is a compact way to name all functions whose derivative equals the integrand. The symbol $dx$ identifies the variable with respect to which the reversal is being performed, which becomes important when formulas contain several letters. The constant $C$ records the vertical information that differentiation erased.

A useful discipline is to read $\int f(x)\,dx$ aloud as "the family of antiderivatives of $f$ with respect to $x$ ." That sentence makes several common mistakes harder to commit. It reminds us that $+C$ belongs in an indefinite integral, that constants involving other fixed parameters remain constants, and that a final answer should differentiate back to the original integrand on the interval where the formula is intended to hold.

Definition

Indefinite integral

The notation

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

means that $F'(x)=f(x)$ . The function $f$ is the integrand, $dx$ names the variable of integration, and $C$ represents an arbitrary constant.

The symbol $\int$ is not a decorative elongated letter. It instructs us to collect the entire family of antiderivatives. Omitting $+C$ changes a family into one member and loses information.

Worked example

A fractional power

\int \sqrt[3]{x^2}\,dx =\int x^{2/3}\,dx =\frac{x^{5/3}}{5/3}+C =\frac35x^{5/3}+C.

The derivative of $\frac35x^{5/3}$ is $x^{2/3}$ , so the result checks.

Common mistake

The exponent $-1$ is the exception

The power rule for antiderivatives,

\int x^n\,dx=\frac{x^{n+1}}{n+1}+C,

requires $n\ne-1$ . When $n=-1$ , the denominator $n+1$ is zero. Instead,

\int \frac1x\,dx=\ln|x|+C.

Interactive checku3a-c-01

Evaluate $\int(4x^{-2}+3x)\,dx$ .

Your work stays on this device. No account or AI grader is used.

Show hint

For $x^{-2}$ , raise the exponent to $-1$ and divide by $-1$ .

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Exercise

Find $\int(7-2x^4)\,dx$ .

Exercise

Find $\int(5/x^3+e^x)\,dx$ .

Exercise

Explain why $\int_0^1 f(x)\,dx$ does not require a $+C$ , while $\int f(x)\,dx$ does.

Exercise

Differentiate your answer to every computational exercise.

After the explanation

Use the section idea

Reading lens

Connect every antiderivative to a derivative check, and every varying rate to a sum of rate-times-width contributions.

Mental model

Indefinite integration recovers a family of functions; definite accumulation combines signed local changes into one net change.

Decision

Ask whether the task wants a general antiderivative, an initial-condition solution, displacement, distance, or a numerical total from data.

Common trap

Omitting the arbitrary constant, confusing displacement with distance, or multiplying one changing rate by the entire interval.

Check yourself

Can you differentiate your antiderivative and interpret the sign and units of a rate-based total?

Source & rights

Original instruction with traceable references.

BetterGrades-original; no direct adaptation declared in the verified handoff.

Reference textbooks remain rights-separated and are not published as application assets. Any direct adaptation requires separate identification and attribution.