\textbf{Linearity property}

Integration is linear. This means the integral can "go through" a sum or a multiplication with a constant.

,,\qquad\int\left(\alpha f(x) + \beta g(x)\right)\mathrm{d}x = \alpha\int f(x)\mathrm{d}x + \beta\int g(x)\mathrm{d}x

,,\qquad\int\left(\sum_{k=1}^n \alpha_k f_k(x)\right)\mathrm{d}x = \sum_{k=1}^n\alpha_k\int f_k(x)\mathrm{d}x

Using this property, it's possible to split an integral into manysmaller integrals which can be solved independently using other methods (e.g. sing table)

\textbf{Example:}

,,\qquad\int\left(\frac{5}{x} + 7\sin(2x) + x^8\right)\mathrm{d}x = 5\int\frac{\mathrm{d}x}{x} + 7\int\sin(2x)\mathrm{d}x + \int x^8\mathrm{d}x = 5\ln|x| -\frac{7}{2}\cos(2x) + \frac{x^9}{9} +C