HSC Maths Advanced Syllabus

What Is Integration?

Integration is a fundamental concept in calculus, and is essentially the inverse operation of differentiation. This means that the derivative of any function, can be reverted back to it’s primitive function (what it was originally) by integrating. Thus, integration is mainly used to find areas, volumes and points of specific functions.

An Integral is denoted by the symbol $\int$, where whatever follows the symbol is referred to as the expression which is being integrated. As such, we use the following notation to integrate a specific function.

$$\int f(x)dx$$

The Definite Integral

Definite integrals are integrals which calculate the accumulation of quantities, such as areas under curves, over a specific interval. For example, a definite integral can be governed by the interval $[a,b]$. This is displayed in integral form as follows:

$$\int_a^bf(x)dx$$

The result of this is a number that represents the total accumulation of what you are finding (Eg. the area under a graph)

Indefinite Integrals

Indefinite integrals, also known as antiderivatives, are used to determine the original function, when given it’s derivative. In simpler terms, after deriving a function, you use an indefinite integral to get back to your original function. This differs from a definite integral, as it is not governed by any limits (this will be explored in a later chapter). This is displayed in integral form as follows:

$$\int f(x)dx$$

What is the Primitive of a Function?

In calculus, a primitive function (also known as an antiderivative) is the function which occurs as a result of integrating a differentiated function. Thus, the primitive of a function is essentially found by integrating a function that has been derived, back to its original form. This concept is fundamental in the study of integral calculus. We will learn more about primitive functions and how they are found throughout this module!