Integralen av $$$x^{2} z^{2} - \frac{3}{2}$$$ med avseende på $$$x$$$

Bestäm $$$\int \left(x^{2} z^{2} - \frac{3}{2}\right)\, dx$$$.

Integrera termvis:

$${\color{red}{\int{\left(x^{2} z^{2} - \frac{3}{2}\right)d x}}} = {\color{red}{\left(- \int{\frac{3}{2} d x} + \int{x^{2} z^{2} d x}\right)}}$$

Tillämpa konstantregeln $$$\int c\, dx = c x$$$ med $$$c=\frac{3}{2}$$$:

$$\int{x^{2} z^{2} d x} - {\color{red}{\int{\frac{3}{2} d x}}} = \int{x^{2} z^{2} d x} - {\color{red}{\left(\frac{3 x}{2}\right)}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=z^{2}$$$ och $$$f{\left(x \right)} = x^{2}$$$:

$$- \frac{3 x}{2} + {\color{red}{\int{x^{2} z^{2} d x}}} = - \frac{3 x}{2} + {\color{red}{z^{2} \int{x^{2} d x}}}$$

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

$$- \frac{3 x}{2} + z^{2} {\color{red}{\int{x^{2} d x}}}=- \frac{3 x}{2} + z^{2} {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=- \frac{3 x}{2} + z^{2} {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

Alltså,

$$\int{\left(x^{2} z^{2} - \frac{3}{2}\right)d x} = \frac{x^{3} z^{2}}{3} - \frac{3 x}{2}$$

Förenkla:

$$\int{\left(x^{2} z^{2} - \frac{3}{2}\right)d x} = \frac{x \left(2 x^{2} z^{2} - 9\right)}{6}$$

Lägg till integrationskonstanten:

$$\int{\left(x^{2} z^{2} - \frac{3}{2}\right)d x} = \frac{x \left(2 x^{2} z^{2} - 9\right)}{6}+C$$

$$$\int \left(x^{2} z^{2} - \frac{3}{2}\right)\, dx = \frac{x \left(2 x^{2} z^{2} - 9\right)}{6} + C$$$A