Integration as an Inverse Process of Differentiation

7.2 Integration as an Inverse Process of Differentiation

The Fundamental Idea

Differentiation gives us the rate at which a function changes. Integration reverses this: we start with the derivative and ask, "What original function could have produced this?" This reverse process is called anti-differentiation or integration.

Consider three familiar derivatives:

In the first case, $\cos x$ is the derivative of $\sin x$. We therefore say that $\sin x$ is an anti-derivative (or an integral) of $\cos x$. Similarly, $\frac{x^3}{3}$ is an anti-derivative of $x^2$, and $e^x$ is an anti-derivative of itself.

The Constant of Integration

Here is a crucial observation: the derivative of any constant function is zero. Therefore, if we add any constant $C$ to $\sin x$, the derivative remains $\cos x$:

Similarly:

This means anti-derivatives are not unique. For any given function, there exist infinitely many anti-derivatives, all differing by a constant. The constant $C$ is called the constant of integration (or an arbitrary constant), and it can be any real number.

Why Functions with the Same Derivative Differ by a Constant

›Proof

Let $g$ and $h$ be two functions that have the same derivative on an interval $I$. Define $f(x) = g(x) - h(x)$ for all $x \in I$.

Differentiating:

$f'(x) = g'(x) - h'(x)$

Since $g'(x) = h'(x)$ by hypothesis:

$f'(x) = 0 \quad \text{for all } x \in I$

This means the rate of change of $f$ with respect to $x$ is zero everywhere on $I$. Therefore $f$ must be constant on $I$. Hence $g(x) - h(x) = C$ for some constant $C$, or $g(x) = h(x) + C$.

This result justifies the statement that $\{F + C : C \in \mathbb{R}\}$ gives all possible anti-derivatives of $f$.

Notation for Indefinite Integrals

We introduce a special symbol to represent the entire family of anti-derivatives:

This is read as "the indefinite integral of $f$ with respect to $x$". The symbol $\int$ is the integral sign, $f(x)$ is the integrand, $x$ is the variable of integration, and $C$ is the constant of integration.

Standard Integrals from Known Derivatives

Since integration reverses differentiation, every derivative formula gives us an integral formula. The following table lists the standard results we will use to find integrals of other functions.