Calculus AB · Integration and Accumulation of Change · 14 min read · Updated 2026-05-10
Antiderivatives and indefinite integrals (basic rules) — AP Calculus AB
AP Calculus AB · Integration and Accumulation of Change · 14 min read
1. Antiderivatives, Indefinite Integrals, and the Constant of Integration★★☆☆☆⏱ 3 min
Because the derivative of any constant is zero, if $F(x)$ is an antiderivative of $f(x)$, then $F(x) + C$ for any real constant $C$ is also an antiderivative. The differential $dx$ in the notation explicitly identifies the variable of integration. If given an initial condition (a point the antiderivative must pass through), you can solve for a specific value of $C$ to get a particular antiderivative.
Exam tip: Always check your antiderivative by differentiating it! If the derivative of your result equals the original integrand, you know your work is correct. This catches 90% of common sign and arithmetic errors on the exam.
2. Algebraic Basic Integration Rules★★☆☆☆⏱ 4 min
Algebraic integration rules directly mirror differentiation rules, reversed for integration. The constant multiple rule states $\int k \cdot f(x) \, dx = k \int f(x) \, dx$ for any constant $k$, and the sum/difference rule states $\int [f(x) \pm g(x)] dx = \int f(x) dx \pm \int g(x) dx$. The most widely used rule is the power rule for integration.
Exam tip: Always rewrite roots and reciprocals as power terms before applying the power rule. Skipping this step is the most common cause of miscalculating the exponent in the power rule on AP exams.
3. Basic Transcendental Antiderivatives (Exponential and Trigonometric)★★★☆☆⏱ 4 min
All rules for non-algebraic functions are direct reverses of derivative rules you already know. The key results for AP Calculus AB are:
Reciprocal ($n=-1$ exception): $\int \frac{1}{x} dx = \ln|x| + C$. The absolute value covers all non-zero $x$.
Trigonometric: $\int \cos x dx = \sin x + C$, $\int \sin x dx = -\cos x + C$, $\int \sec^2 x dx = \tan x + C$, $\int \sec x \tan x dx = \sec x + C$
Exam tip: If you are unsure about the sign of a trigonometric antiderivative, take 10 seconds to differentiate your result to confirm it matches the original integrand. FRQ graders deduct points for incorrect signs, so this check is well worth the time.
4. AP-Style Concept Check Practice★★★☆☆⏱ 3 min
Common Pitfalls
Why: Students blindly apply the power rule without remembering the exception for $n=-1$.
Why: Students confuse the original function with its antiderivative.
Why: Students memorize the rule without remembering the domain of $\frac{1}{x}$ includes negative $x$.
Why: Students apply the basic power rule directly to composite functions before learning u-substitution.
Why: Students think each term needs its own constant.
Why: Students mix up the power rule for polynomials with the exponential integration rule.