Indefinite Integrals and Net Change
This concept page explores the relationship between antiderivatives, indefinite integrals, and the net change theorem.
Indefinite Integrals
Definition
The indefinite integral of $f(x)$ is:
$$\int f(x)\,dx = F(x) + C$$
where $F$ is any antiderivative of $f$ (meaning $F'(x) = f(x)$), and $C$ is an arbitrary constant.
Key Properties
Linearity: $\int [af(x) + bg(x)]\,dx = a\int f(x)\,dx + b\int g(x)\,dx$
Power Rule: $\int x^n\,dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)$
Special Cases:
- $\int e^x\,dx = e^x + C$
- $\int \frac{1}{x}\,dx = \ln\vert x\vert + C$
- $\int \cos x\,dx = \sin x + C$
- $\int \sin x\,dx = -\cos x + C$
The Net Change Theorem
Statement
If $F'(x) = f(x)$ is continuous on $[a, b]$, then:
$$\int_a^b f(x)\,dx = F(b) - F(a)$$
This is the Fundamental Theorem of Calculus, Part 2.
Interpretation
The definite integral $\int_a^b f(x)\,dx$ gives the net change in $F$ over the interval $[a, b]$.
| If $f(x)$ represents... | Then $\int_a^b f(x)\,dx$ gives... |
|---|---|
| Velocity $v(t)$ | Displacement (net change in position) |
| Rate of population change $P'(t)$ | Net change in population |
| Marginal cost $C'(x)$ | Net change in total cost |
Displacement vs. Total Distance
Displacement (Net Change)
$$\text{Displacement} = \int_a^b v(t)\,dt$$
This can be positive, negative, or zero.
Total Distance
$$\text{Total Distance} = \int_a^b \vert v(t)\vert \,dt$$
This is always non-negative and accounts for all movement.
Example: If you walk 5 m east then 3 m west:
- Displacement = 2 m (east)
- Total distance = 8 m
Common Misconceptions
the integral of a product equals the product of the integrals.
This is the multiplicative-not-additive error. In general, $\int f(x)g(x)\,dx \neq \int f(x)\,dx \cdot \int g(x)\,dx$. A direct counterexample: $\int x \cdot x\,dx = \int x^2\,dx = \frac{x^3}{3} + C$, whereas $\int x\,dx \cdot \int x\,dx = \frac{x^2}{2} \cdot \frac{x^2}{2} = \frac{x^4}{4}$, which is a different function. Integration distributes over addition (linearity) but not over multiplication.
the constant of integration $+ C$ is a formality that can be omitted.
This is the action-view-of-function error. The $+ C$ encodes a genuine family of functions, each shifted vertically. Since differentiation destroys constant information, the inverse process recovers not a single function but an entire class. In differential equations, the $+ C$ is precisely the free parameter that an initial condition pins down. Omitting it produces an incomplete answer that cannot satisfy arbitrary initial conditions.
Related Skills
- Indefinite Integral Notation
- Computing Indefinite Integrals
- Net Change Theorem
- Displacement vs. Total Distance