Integralen av $$$-2 + \frac{1}{u^{2}}$$$

Bestäm $$$\int \left(-2 + \frac{1}{u^{2}}\right)\, du$$$.

Integrera termvis:

$${\color{red}{\int{\left(-2 + \frac{1}{u^{2}}\right)d u}}} = {\color{red}{\left(- \int{2 d u} + \int{\frac{1}{u^{2}} d u}\right)}}$$

Tillämpa konstantregeln $$$\int c\, du = c u$$$ med $$$c=2$$$:

$$\int{\frac{1}{u^{2}} d u} - {\color{red}{\int{2 d u}}} = \int{\frac{1}{u^{2}} d u} - {\color{red}{\left(2 u\right)}}$$

Tillämpa potensregeln $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=-2$$$:

$$- 2 u + {\color{red}{\int{\frac{1}{u^{2}} d u}}}=- 2 u + {\color{red}{\int{u^{-2} d u}}}=- 2 u + {\color{red}{\frac{u^{-2 + 1}}{-2 + 1}}}=- 2 u + {\color{red}{\left(- u^{-1}\right)}}=- 2 u + {\color{red}{\left(- \frac{1}{u}\right)}}$$

Alltså,

$$\int{\left(-2 + \frac{1}{u^{2}}\right)d u} = - 2 u - \frac{1}{u}$$

Lägg till integrationskonstanten:

$$\int{\left(-2 + \frac{1}{u^{2}}\right)d u} = - 2 u - \frac{1}{u}+C$$

$$$\int \left(-2 + \frac{1}{u^{2}}\right)\, du = \left(- 2 u - \frac{1}{u}\right) + C$$$A