Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation
Reverse familiar derivative rules to find antiderivative families on their valid intervals.
College Board Topic 6.8 · CED p. 120 (PDF p. 125) · CED effective Fall 2020; project course outline 2026–27.
An indefinite integral names a family of antiderivatives: ∫f(x)dx=F(x)+C. Reverse familiar derivative rules: ∫xⁿdx=xⁿ⁺¹/(n+1)+C for n≠−1; ∫1/x dx=ln|x|+C on intervals excluding zero; ∫eˣdx=eˣ+C; ∫cos x dx=sin x+C; ∫sin x dx=−cos x+C. Constants and sums pass through the integral.
Also ∫sec²x dx=tan x+C, ∫1/(1+x²)dx=arctan x+C, and ∫1/√(1−x²)dx=arcsin x+C on the appropriate domains. Check signs and chain factors by differentiating. Not every continuous integrand, such as e^(−x²), has an elementary closed form. It can still define an accumulation function or be integrated numerically; failing to find a simple formula does not mean its integral does not exist.
Let's work through it
- Integrate 3x²−2cos x+1/x term by term. Reason: Linearity permits separate antiderivatives.
- Obtain x³−2sin x+ln|x|+C, for x≠0 on an interval. Reason: Reverse the power, cosine, and logarithmic derivative rules.
- Differentiate to recover 3x²−2cos x+1/x. Reason: Verification checks both sign and domain.
Try it from memory
Pause & retrieve
Try it before you peek.
Write your answer and the reason from memory. Then compare with the worked answer. Your self-assessment is a study decision, not an automatically graded score.
Question 1
What goes wrong with the power rule at n=−1?
See every recall question with its answer
These references preserve each prompt and its answer for direct evidence links. Try the interactive questions above before opening the worked reasoning.
Retrieval 1
What goes wrong with the power rule at n=−1?
Worked answer and reason
The denominator n+1 is zero; use ln|x|.
The logarithm has derivative 1/x.
Retrieval 2
Does “no elementary antiderivative” mean “no definite integral”?
Worked answer and reason
No.
Continuity still guarantees an integral; numerical methods or integral notation can describe it.
Catch the tempting mistake
- Every indefinite integral has one numerical answer.
- It is a family of functions, differing by a constant on each interval of definition.
Calculator workspace and limits
For calculus, set the angle selector to Radians. This scientific calculator evaluates arithmetic and functions. It does not graph arbitrary functions, find numerical roots, evaluate numerical derivatives, or compute definite integrals. Those tasks need an approved graphing tool; the May 2027 exam permits approved handheld graphing calculators and Bluebook’s built-in Desmos on calculator-required parts. A graph or decimal estimate helps you check a conclusion; your setup, hypotheses, units and justification still do the explaining.
Scientific calculator
This calculator requires JavaScript. Use the physics formula sheets and show your working manually.
+ − * / ^ ! · pi · e · sqrt · sin, cos, tan · asin, acos, atan · atan2(y,x) · ln, log · abs · exp · hypot(x,y). Use * between factors. Keep units in your written working.
Put it to work
- Unit 6 problem 19 · No calculator · analytic
- Unit 6 problem 20 · No calculator · analytic
- Unit 6 problem 26 · No calculator · verbal
For the teacher
- Include reverse-rule retrieval before mixed practice. Ask students to check one answer by differentiation, rather than trusting a remembered pattern.
Glossary and deeper reading · 4 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
-
Indefinite integral
An indefinite integral denotes the family of antiderivatives of its integrand on an interval. The constant of integration records the possible constant differences among those functions. For example, the indefinite integral of cos x is sin x+C. It has no pair of numerical bounds and does not by itself give a single accumulated amount.
-
Antiderivative
An antiderivative of a function is another function whose derivative equals the original function on an interval. Two antiderivatives on the same interval differ by a constant. For example, x² is an antiderivative of 2x, and x²+7 is another. An antiderivative is a function, whereas a definite integral is an accumulated value.
-
Domain
The domain of a function is the set of inputs for which the function is defined. An expression and its context can both restrict the allowed inputs. For example, the real function f(x)=1/(x−2) has domain all real numbers except 2. For a model of elapsed time, a sensible domain may also require t≥0.
-
Integrand
The integrand is the function or expression being integrated. The differential identifies the variable of integration. For example, in the integral of x² with respect to x, the integrand is x². Bounds belong to the integral’s interval rather than to the integrand itself.