Integralen av $$$- \frac{2 x}{\pi} + 1$$$

Bestäm $$$\int \left(- \frac{2 x}{\pi} + 1\right)\, dx$$$.

Integrera termvis:

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

Tillämpa konstantregeln $$$\int c\, dx = c x$$$ med $$$c=1$$$:

$$- \int{\frac{2 x}{\pi} d x} + {\color{red}{\int{1 d x}}} = - \int{\frac{2 x}{\pi} d x} + {\color{red}{x}}$$

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

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

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

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

Alltså,

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

Förenkla:

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

Lägg till integrationskonstanten:

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

$$$\int \left(- \frac{2 x}{\pi} + 1\right)\, dx = \frac{x \left(\pi - x\right)}{\pi} + C$$$A