Integralen av $$$5 \sin{\left(5 x \right)}$$$

Bestäm $$$\int 5 \sin{\left(5 x \right)}\, dx$$$.

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=5$$$ och $$$f{\left(x \right)} = \sin{\left(5 x \right)}$$$:

$${\color{red}{\int{5 \sin{\left(5 x \right)} d x}}} = {\color{red}{\left(5 \int{\sin{\left(5 x \right)} d x}\right)}}$$

Låt $$$u=5 x$$$ vara.

Då $$$du=\left(5 x\right)^{\prime }dx = 5 dx$$$ (stegen kan ses »), och vi har att $$$dx = \frac{du}{5}$$$.

Alltså,

$$5 {\color{red}{\int{\sin{\left(5 x \right)} d x}}} = 5 {\color{red}{\int{\frac{\sin{\left(u \right)}}{5} d u}}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ med $$$c=\frac{1}{5}$$$ och $$$f{\left(u \right)} = \sin{\left(u \right)}$$$:

$$5 {\color{red}{\int{\frac{\sin{\left(u \right)}}{5} d u}}} = 5 {\color{red}{\left(\frac{\int{\sin{\left(u \right)} d u}}{5}\right)}}$$

Integralen av sinus är $$$\int{\sin{\left(u \right)} d u} = - \cos{\left(u \right)}$$$:

$${\color{red}{\int{\sin{\left(u \right)} d u}}} = {\color{red}{\left(- \cos{\left(u \right)}\right)}}$$

Kom ihåg att $$$u=5 x$$$:

$$- \cos{\left({\color{red}{u}} \right)} = - \cos{\left({\color{red}{\left(5 x\right)}} \right)}$$

Alltså,

$$\int{5 \sin{\left(5 x \right)} d x} = - \cos{\left(5 x \right)}$$

Lägg till integrationskonstanten:

$$\int{5 \sin{\left(5 x \right)} d x} = - \cos{\left(5 x \right)}+C$$

$$$\int 5 \sin{\left(5 x \right)}\, dx = - \cos{\left(5 x \right)} + C$$$A