Integration

If we were to look at a function on a graph, the integral of would describe the area underneath the function. To represent the integral of , we would write , with telling us we are integrating with respect to x. (This is called the differential).

Conceptually, we think of integration as being the inverse of differentiation. This means that to find an integral, we can think of having to 'undo' the process of differentiation. When we have 'undone' the integration, we call this result the antiderivative.

When we find an indefinite integral, it is important that we add a constant of integration, meaning that if we were to find , we would give our answer as . This reflects that this antiderivative function could have any constant and still differentiated to the original function.

Find

differentiates to , so that is our antiderivative. However, in full, our answer is , as we must include this constant of integration.

Visualisation of a definite integral

Visualisation of a definite integral

The area of the strips at this point is given as . To find the value of the integral, we need to use an infinite number of strips to cover the inside of the curve fully. This means as we take the limit, we get .

• For constants and , and functions , then

  

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• for

By the product rule (as seen in differentiation), for two functions and then

which then simplifies to -

There is also an option to use substitution to simplify an integral. Here we change the variable we integrate with respect to. In the case of a definite integral, the limits also need to be changed using the substitution. We must also change the integrand. This is best demonstrated by an example. Trying to choose the right substitution takes time. However, it becomes easier.

and , then the integral of the curve defined by these functions is given as . We can think of this as the 's canceling to give , which is what we'd expect in a normal integral.