Integration as Anti-Differentiation
Integration serves as the inverse operation of differentiation, otherwise known as anti-differentiation (or finding the anti-derivative).
Polynomial Anti-Differentiation
The focus is on functions of the form $f(x) = ax^n + bx^{n-1} + ...$, where $n$ is an real number and $n \neq -1$. To anti-differentiate such functions, we apply the following rule:
$$\int x^n dx = \frac{x^{n+1}}{n+1} + C, \text{ where } n \neq -1$$
Here, $C$ represents the constant of integration, which accounts for the fact that the derivative of a constant is zero.
ExampleConsider the function $f(x) = 3x^2 + 2x - 5$. Its anti-derivative is:
$\int (3x^2 + 2x - 5) dx = x^3 + x^2 - 5x + C$
To verify, we can differentiate the result:
$\frac{d}{dx}(x^3 + x^2 - 5x + C) = 3x^2 + 2x - 5$
NoteThe process of anti-differentiation is not unique without additional information, as different constants of integration will yield different but equally valid anti-derivatives.
Boundary Conditions and the Constant Term
To determine the specific anti-derivative that satisfies a given condition, we use boundary conditions. This process involves finding the value of the constant $C$ that makes the anti-derivative satisfy a given point.
ExampleFind the anti-derivative $F(x)$ of $f(x) = 2x + 3$ that passes through the point (1, 5).
First, we anti-differentiate: $F(x) = \int (2x + 3) dx = x^2 + 3x + C$
Now, we use the boundary condition $F(1) = 5$: $5 = 1^2 + 3(1) + C$ $5 = 1 + 3 + C$ $C = 1$
Therefore, the specific anti-derivative is $F(x) = x^2 + 3x + 1$.
Definite Integrals and Technology
A definite integral represents the signed area between a function and the $x$-axis over a specified interval (meaning if the area is below the $x$-axis, it would be counted as negative. Students are expected to use technology to evaluate definite integrals.
The definite integral of a function $f(x)$ from $a$ to $b$ is denoted as:
$$\int_a^b f(x) dx$$
