Integration as the Inverse of Differentiation

3.1

Integration as the Inverse of Differentiation

Definition

Integration is the reverse process of differentiation.

Notation

An integration process is denoted by the symbol $\int ... \, dx$.

$The image illustrates the relationship between differentiation and integration in calculus. It shows two main processes: 1. Differentiation: Represented at the top with the formula $ \frac{d}{dx} [f(x)] = f'(x) $. This process converts the function $ f(x) $ into its derivative $ f'(x) $. 2. Integration: Represented at the bottom with the formula $ \int f'(x) \, dx = f(x) $. This process converts the derivative $ f'(x) $ back into the original function $ f(x) $. Arrows indicate the direction of these processes, showing that differentiation and integration are inverse operations. The Pandai logo is present in the center$

Integration

If $\dfrac{d}{dx}[f(x)]=f'(x)$, then the integral of $f'(x)$ with respect to $x$ is $\int f'(x) \, dx=f(x)$.

Example

Question

Given $\dfrac{d}{dx}(4x^2)=8x$, find $\int 8x \, dx$.

Solution

Differentiation of $4x^2$ is $8x$.

By the reverse of differentiation, the integration of $8x$ is $4x^2$.

Hence, $\int 8x \, dx=4x^2$.