Properties of Integrals

Integration behaves linearly with respect to constant multiples and sums/differences of functions — this is what lets a complicated-looking integrand be broken into manageable pieces and integrated term by term.

Note 11.1 — combining and extending both properties. The two properties above combine, and extend to any finite number of terms, as

In plain words: the integral of a linear combination of finitely many functions equals the same linear combination of their individual integrals. This is the justification for the routine step, used constantly from here on, of integrating a multi-term expression one term at a time and simply adding up the results (each carrying its own working constant, which are then merged into one overall arbitrary constant $c$).

Worked illustration (pure decomposition). To integrate $2\cos x - 4\sin x + 5\sec^2x + \text{cosec}^2x$, split into four separate standard integrals and combine:

(Each coefficient is pulled out by Property 1, and each piece read off directly from the standard-integrals table of §11.3.)

Worked illustration (mixing decomposition with the linear-argument rule of §11.4). A realistic problem typically needs both properties of this section and the $\int f(ax+b)dx$ shortcut together. For instance,

and each of the three pieces is then evaluated by §11.4's rule before being recombined:

Similarly, a mixture of a reciprocal-square-root term (§11.4 inverse-trig case) with a cosec-cot product term decomposes the same way — each term integrated on its own by whichever standard/linear-argument rule applies, then the results simply added with their original signs: …