All the general theorems of § 113 may of course also be stated as theorems in integration. Thus we have, to begin with, the formulae \[\begin{equation*} \int \{f(x) + F(x)\}\, dx = \int f(x) dx + \int F(x) dx, \tag{1}\end{equation*}\] \[\begin{equation*} \int kf(x)\, dx = k\int f(x) dx. \tag{2} \end{equation*}\]
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 \[\int (a_{0}x^{n} + a_{1}x^{n-1} + \dots + a_{n})\, dx = \frac{a_{0}x^{n+1}}{n + 1} + \frac{a_{1}x^{n}}{n} + \dots + a_{n}x.\]
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