Antiderivatives and indefinite integrals (basic rules) — AP Calculus BC
1. Core Definitions and Notation ★☆☆☆☆ ⏱ 3 min
An antiderivative of a function $f(x)$ is a differentiable function $F(x)$ such that $F'(x) = f(x)$ for all $x$ in the domain of $f$. The indefinite integral of $f(x)$ is the general form of all antiderivatives of $f$, which form a family of functions differing only by a vertical shift.
This topic is the foundational starting point for all integration work in AP Calculus BC, making up 1-3% of total exam score. It acts as a building block for all larger problems including differential equations, area/volume calculations, and accumulation of change.
2. Core Algebraic Antiderivative Rules ★★☆☆☆ ⏱ 4 min
All basic antiderivative rules are derived by reversing corresponding derivative rules. The constant multiple and sum/difference rules allow us to pull constants out of integrals and integrate term-by-term:
\int k f(x) dx = k \int f(x) dx
\int \left(f(x) \pm g(x)\right) dx = \int f(x) dx \pm \int g(x) dx
The power rule for integration reverses the power rule for derivatives, with a critical exception for $n=-1$:
\int x^n dx = \frac{x^{n+1}}{n+1} + C, \quad n \neq -1
When $n=-1$, the integrand is $\frac{1}{x}$, and the power rule would result in division by zero. For this case, the antiderivative is $\ln|x| + C$, since the derivative of $\ln|x|$ is $\frac{1}{x}$ for all non-zero $x$.
Exam tip: Always rewrite roots and rational terms as $x^n$ before applying the power rule to avoid mistakes with negative and fractional exponents.
3. Exponential and Logarithmic Antiderivative Rules ★★☆☆☆ ⏱ 3 min
Reversing derivative rules for exponential and logarithmic functions gives core rules frequently used in applied calculus problems. The simplest rule is for the natural exponential function, which is its own antiderivative:
\int e^x dx = e^x + C
For general exponential functions with constant base $a \neq e$, we reverse the derivative rule $\frac{d}{dx}[a^x] = a^x \ln a$ by dividing by $\ln a$:
\int a^x dx = \frac{a^x}{\ln a} + C, \quad a>0, a \neq 1
For the natural logarithm itself, the antiderivative is derived from the product rule for derivatives:
\int \ln x dx = x \ln x - x + C, \quad x>0
Exam tip: Always check if your function is a power function or exponential function before integrating; if the variable is in the exponent, use the exponential rule, not the power rule.
4. Trigonometric Antiderivative Rules ★★★☆☆ ⏱ 4 min
All basic trigonometric antiderivative rules are derived directly from reversing derivative rules, and these are not provided on the AP Calculus formula sheet, so memorization is required:
- $\int \cos x dx = \sin x + C$
- $\int \sin x dx = -\cos x + C$
- $\int \sec^2 x dx = \tan x + C$
- $\int \csc^2 x dx = -\cot x + C$
- $\int \sec x \tan x dx = \sec x + C$
- $\int \csc x \cot x dx = -\csc x + C$
Exam tip: Differentiate your final antiderivative to catch sign errors before moving on in a problem.
Common Pitfalls
Why: Students automatically apply the power rule without remembering the $n \neq -1$ exception.
Why: Confusion between exponential functions (variable exponent, constant base) and power functions (constant exponent, variable base).
Why: Students forget that indefinite integrals require the arbitrary constant, especially when carrying the result into a later part of a problem.
Why: Students memorize only the derivative rule and forget to reverse the sign correctly when integrating.
Why: Students make an algebra mistake when simplifying the fraction after applying the power rule.
Why: Students only work with positive $x$ in examples and forget that $\frac{1}{x}$ is defined for negative $x$, and $\ln x$ is not.