# Integrating 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 $y=6x^2-3x^5$.

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