Integration as the inverse of differentiation

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