Integralen av $$$- \frac{\pi^{\pi} \sin{\left(x \right)}}{x}$$$

Bestäm $$$\int \left(- \frac{\pi^{\pi} \sin{\left(x \right)}}{x}\right)\, dx$$$.

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

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

Denna integral (Sinusintegralen) har ingen sluten form:

$$- \pi^{\pi} {\color{red}{\int{\frac{\sin{\left(x \right)}}{x} d x}}} = - \pi^{\pi} {\color{red}{\operatorname{Si}{\left(x \right)}}}$$

Alltså,

$$\int{\left(- \frac{\pi^{\pi} \sin{\left(x \right)}}{x}\right)d x} = - \pi^{\pi} \operatorname{Si}{\left(x \right)}$$

Lägg till integrationskonstanten:

$$\int{\left(- \frac{\pi^{\pi} \sin{\left(x \right)}}{x}\right)d x} = - \pi^{\pi} \operatorname{Si}{\left(x \right)}+C$$

$$$\int \left(- \frac{\pi^{\pi} \sin{\left(x \right)}}{x}\right)\, dx = - \pi^{\pi} \operatorname{Si}{\left(x \right)} + C$$$A