Integration is the reverse of differentiation.

If you take a function, differentiate it and then integrate the resulting expression, you will be left with the original function.

Similarly, if you integrate something and then differentiate it, you will also be left the original function.

While the derivative can be said to represent the gradient of the curve, the integral represents the area underneath the curve.

There are two types of integral - definite integrals and indefinite integrals.

A definite integral has upper and lower bounds defined. It gives an expression for the area underneath the curve between the upper and lower limits.

In indefinite integral does not have its upper and lower bounds defined. Because of this, it must contain an arbitrary constant, \(C\). The diagram below demonstrates why this constant is needed. All three of the lines have the same gradient - differentiating them would give the same answer. However, there is not the same area under each of the curves( Area here referring to the area between the curve and the x axis). Clearly, the curve above the red section has a much greater area than the curve above the yellow section.

Indefinite Integrals