Integralen av $$$- \frac{x^{2}}{79} + \frac{76823 x}{79}$$$

Bestäm $$$\int \left(- \frac{x^{2}}{79} + \frac{76823 x}{79}\right)\, dx$$$.

Integrera termvis:

$${\color{red}{\int{\left(- \frac{x^{2}}{79} + \frac{76823 x}{79}\right)d x}}} = {\color{red}{\left(\int{\frac{76823 x}{79} d x} - \int{\frac{x^{2}}{79} d x}\right)}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=\frac{1}{79}$$$ och $$$f{\left(x \right)} = x^{2}$$$:

$$\int{\frac{76823 x}{79} d x} - {\color{red}{\int{\frac{x^{2}}{79} d x}}} = \int{\frac{76823 x}{79} d x} - {\color{red}{\left(\frac{\int{x^{2} d x}}{79}\right)}}$$

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

$$\int{\frac{76823 x}{79} d x} - \frac{{\color{red}{\int{x^{2} d x}}}}{79}=\int{\frac{76823 x}{79} d x} - \frac{{\color{red}{\frac{x^{1 + 2}}{1 + 2}}}}{79}=\int{\frac{76823 x}{79} d x} - \frac{{\color{red}{\left(\frac{x^{3}}{3}\right)}}}{79}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=\frac{76823}{79}$$$ och $$$f{\left(x \right)} = x$$$:

$$- \frac{x^{3}}{237} + {\color{red}{\int{\frac{76823 x}{79} d x}}} = - \frac{x^{3}}{237} + {\color{red}{\left(\frac{76823 \int{x d x}}{79}\right)}}$$

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

$$- \frac{x^{3}}{237} + \frac{76823 {\color{red}{\int{x d x}}}}{79}=- \frac{x^{3}}{237} + \frac{76823 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}}{79}=- \frac{x^{3}}{237} + \frac{76823 {\color{red}{\left(\frac{x^{2}}{2}\right)}}}{79}$$

Alltså,

$$\int{\left(- \frac{x^{2}}{79} + \frac{76823 x}{79}\right)d x} = - \frac{x^{3}}{237} + \frac{76823 x^{2}}{158}$$

Förenkla:

$$\int{\left(- \frac{x^{2}}{79} + \frac{76823 x}{79}\right)d x} = \frac{x^{2} \left(230469 - 2 x\right)}{474}$$

Lägg till integrationskonstanten:

$$\int{\left(- \frac{x^{2}}{79} + \frac{76823 x}{79}\right)d x} = \frac{x^{2} \left(230469 - 2 x\right)}{474}+C$$

$$$\int \left(- \frac{x^{2}}{79} + \frac{76823 x}{79}\right)\, dx = \frac{x^{2} \left(230469 - 2 x\right)}{474} + C$$$A