Antiderivatives and the Fundamental Theorem of Calculus

Antiderivatives

Definition of Antiderivative

An antiderivative of a function f on an interval I is a function F such that the derivative of F is f for all x in I:

$F'(x) = f(x)$ for all $x$ in $I$.

General Antiderivative

If F is an antiderivative of f on an interval I, then the most general antiderivative of f on I is:

$F(x) + C$

where C is an arbitrary constant. This means all antiderivatives of f differ by a constant.

Family of Antiderivatives

Assigning different values to C produces a family of functions, each a vertical translation of the others. For example, the general antiderivative of $f(x) = x^2$ is $\frac{1}{3}x^3 + C$.

Common Antiderivative Formulas

Some frequently used antiderivative formulas are summarized in the table below. These include power, exponential, logarithmic, and trigonometric functions.

Function	General Antiderivative
$x^n$	$\frac{1}{n+1}x^{n+1} + C$, $n \neq -1$
$\sin kx$	$-\frac{1}{k}\cos kx + C$
$\cos kx$	$\frac{1}{k}\sin kx + C$
$e^{kx}$	$\frac{1}{k}e^{kx} + C$
$\frac{1}{x}$	$\ln|x| + C$, $x \neq 0$

Linearity of Antiderivatives

Antiderivatives are linear, meaning:

• Constant Multiple Rule: $\int kf(x)\,dx = k\int f(x)\,dx$

• Sum or Difference Rule: $\int [f(x) \pm g(x)]\,dx = \int f(x)\,dx \pm \int g(x)\,dx$

Indefinite Integral

The collection of all antiderivatives of f is called the indefinite integral of f with respect to x:

$\int f(x)\,dx$

Here, $\int$ is the integral sign, f is the integrand, and x is the variable of integration.

The Fundamental Theorem of Calculus (FTC)

FTC Part 1: Differentiation and Integration as Inverse Processes

If f is continuous on $[a, b]$, then the function $F(x) = \int_a^x f(t)\,dt$ is continuous on $[a, b]$, differentiable on $(a, b)$, and its derivative is $f(x)$:

$F'(x) = \frac{d}{dx}\int_a^x f(t)\,dt = f(x)$

FTC Part 2: Evaluating Definite Integrals

If f is continuous on $[a, b]$ and F is any antiderivative of f on $[a, b]$, then:

$\int_a^b f(x)\,dx = F(b) - F(a)$

Applications of the FTC

• Area under a curve: The definite integral $\int_a^b f(x)\,dx$ gives the net area between the curve $y = f(x)$ and the $x$-axis from $x = a$ to $x = b$.

• Net Change Theorem: The net change in a differentiable function $F(x)$ over $[a, b]$ is the integral of its rate of change:

$F(b) - F(a) = \int_a^b F'(x)\,dx$

Worked Examples and Notes

• Example: Find $\int_1^3 x^2\,dx$. An antiderivative is $F(x) = \frac{1}{3}x^3$, so $F(3) - F(1) = \frac{1}{3}(27) - \frac{1}{3}(1) = 9 - \frac{1}{3} = \frac{26}{3}$.

• Example: Find $\int_3^6 \frac{1}{x}\,dx$. An antiderivative is $F(x) = \ln|x|$, so $F(6) - F(3) = \ln 6 - \ln 3 = \ln 2$.

• Note: The FTC applies only to continuous functions on $[a, b]$. If $f$ is not continuous (e.g., $f(x) = 1/x^2$ on $[-1, 3]$), the definite integral may not exist.

Inverse Processes

• FTC1: Differentiating the integral of $f$ returns $f$.

• FTC2: Integrating the derivative of $F$ returns $F(b) - F(a)$.

Handwritten Example Summary

The handwritten notes illustrate the application of the FTC to compute derivatives of integrals with variable limits, and the use of the Net Change Theorem. For example, $\frac{d}{dx} \int_0^{x^3} \sin(5t)\,dt = \sin(5x^3) \cdot 3x^2$ by the Chain Rule and FTC1.