129. Integration of polynomials

All the general theorems of § 113 may of course also be stated as theorems in integration. Thus we have, to begin with, the formulae $$\begin{array}{}\text{(1)}& \int \{f(x)+F(x)\}dx=\int f(x)dx+\int F(x)dx,\end{array}$$ $$\begin{array}{}\text{(2)}& \int kf(x)dx=k\int f(x)dx.\end{array}$$

Here it is assumed, of course, that the arbitrary constants are adjusted properly. Thus the formula (1) asserts that the sum of any integral of $f(x)$ and any integral of $F(x)$ is an integral of $f(x)+F(x)$.

These theorems enable us to write down at once the integral of any function of the form $\sum A_{\nu }{f}_{\nu }(x)$, the sum of a finite number of constant multiples of functions whose integrals are known. In particular we can write down the integral of any polynomial: thus $$\int (a_0{x}^n+a_1{x}^{n-1}+\cdots +a_n)dx=\frac{a_0{x}^{n+1}}{n+1}+\frac{a_1{x}^n}{n}+\cdots +a_{n}x.$$