# Integration as Reverse of Differentiation

## Introduction

• Integration is viewed as the reverse process of differentiation in calculus.
• There are two main types of integration: indefinite integration and definite integration.
• The result of integration is labelled as the integral.

## Technique

• Differentiation takes a function and provides its derivative, a function that tells the rate of change.
• The integral of a derivative gives back the original function, showing the reversal aspect of integration.
• For instance, if the derivative of a function F(x) is f(x), then the indefinite integral of f(x) with respect to x gives us F(x) plus a constant, C.

## Notation

• The usual notation for integration is the elongated ‘S’-like symbol ∫ followed by the function we want to integrate and the variable of integration such as dx. For example, ∫f(x) dx.
• The added constant in indefinite integration is often referred to as the constant of integration.
• The ‘a’ and ‘b’ at the top and bottom of the ∫ in definite integration are the limits of integration, indicating the interval over which we are integrating the function.

## Properties of Integration

• Knowledge of the basic integration formulae is essential, usually derived from the standard differentiation formulae.
• The integral of a sum or difference of functions is equal to the sum or difference of their integrals respectively.
• If a function is multiplied by a constant, the integral of the function is also multiplied by the same constant.
• In order to integrate a function made by multiplying two others, techniques like the integration by parts or the use of identities are required.
• The concept of substitution often simplifies integrations. It involves a change of variable to make the integration accessible.

## Further Application of Integration

• Integration is applied in various scenarios including solving ordinary differential equations, calculating areas, volumes and also in physics to understand notions of work and energy.
• Definite integration is used when calculating the area under a curve between two points.
• It also allows us to compute average values of functions.
• Use of definite integration techniques permits the calculation of arc lengths, surface area and volumes of revolution.

## Common Mistakes in Integration

• Forgetting to include the constant of integration in the indefinite integrations.
• Neglecting to apply the basic rules correctly, for example, the power rule.
• Troubles in the application of appropriate methodologies in cases of product or quotients of functions.

## Relevance of Integration

• Serves as a critical tool in understanding and depicting continuous change.
• Holds a significant place in nearly all areas of mathematics, natural sciences and engineering.
• Understanding how the process of integration reverses differentiation is fundamental to calculus and numeric analysis.