A-Level MathsIntegrating Polynomials


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 x^3, requires first multiplying down by the power then reducing the power by one. If y=x^3, \frac{dy}{dx}=3x^2. Integration is the reverse process. Given \frac{dy}{dx}=3x^2, one can find y by integrating, i.e, increasing the power by 1 then dividing by the new power: y=\frac{3x^3}{3}=x^3.

 

Given any y, we can write its integral as

\int y\,dx

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.

\int x^n\, dx=\frac{1}{n+1}x^{n+1}+c

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.


 

Example – Integrate y=6x^2-3x^5.

\int y\, dx=\int\left(6x^2-3x^5\right)dx=\frac{6}{3}x^3-\frac{3}{6}x^6+c=2x^3-\frac{1}{2}x^6+c