Integration can be thought of as the opposite of differentiation, although the fundamental theorem of calculus
isn’t quite this simple.
Differentiating a polynomial term, such as , requires first multiplying down by the power then reducing the power by one. If , . Integration is the reverse process. Given , one can find y by integrating, i.e, increasing the power by 1 then dividing by the new power: .
Given any y, we can write its integral as
This is an indefinite integral, definite integrals also have limits. Integrating a power of x requires adding one to the power then dividing by the new power, i.e.
Notice that we have added c on the end – this is known as the integration constant. The reason this is required whenever integration is performed is because integration is the reverse process of differentiation – whenever a constant is differentiated it disappears. This constant must be brought back in when the reverse process is performed. We call it c conventionally; it represents any constant value.