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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