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

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

För integralen $$$\int{x \cos{\left(5 x \right)} d x}$$$, använd partiell integration $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Låt $$$\operatorname{u}=x$$$ och $$$\operatorname{dv}=\cos{\left(5 x \right)} dx$$$.

Då gäller $$$\operatorname{du}=\left(x\right)^{\prime }dx=1 dx$$$ (stegen kan ses ») och $$$\operatorname{v}=\int{\cos{\left(5 x \right)} d x}=\frac{\sin{\left(5 x \right)}}{5}$$$ (stegen kan ses »).

Integralen kan omskrivas som

$${\color{red}{\int{x \cos{\left(5 x \right)} d x}}}={\color{red}{\left(x \cdot \frac{\sin{\left(5 x \right)}}{5}-\int{\frac{\sin{\left(5 x \right)}}{5} \cdot 1 d x}\right)}}={\color{red}{\left(\frac{x \sin{\left(5 x \right)}}{5} - \int{\frac{\sin{\left(5 x \right)}}{5} d x}\right)}}$$

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

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

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,

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

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

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

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

$$\frac{x \sin{\left(5 x \right)}}{5} - \frac{{\color{red}{\int{\sin{\left(u \right)} d u}}}}{25} = \frac{x \sin{\left(5 x \right)}}{5} - \frac{{\color{red}{\left(- \cos{\left(u \right)}\right)}}}{25}$$

Kom ihåg att $$$u=5 x$$$:

$$\frac{x \sin{\left(5 x \right)}}{5} + \frac{\cos{\left({\color{red}{u}} \right)}}{25} = \frac{x \sin{\left(5 x \right)}}{5} + \frac{\cos{\left({\color{red}{\left(5 x\right)}} \right)}}{25}$$

Alltså,

$$\int{x \cos{\left(5 x \right)} d x} = \frac{x \sin{\left(5 x \right)}}{5} + \frac{\cos{\left(5 x \right)}}{25}$$

Lägg till integrationskonstanten:

$$\int{x \cos{\left(5 x \right)} d x} = \frac{x \sin{\left(5 x \right)}}{5} + \frac{\cos{\left(5 x \right)}}{25}+C$$

$$$\int x \cos{\left(5 x \right)}\, dx = \left(\frac{x \sin{\left(5 x \right)}}{5} + \frac{\cos{\left(5 x \right)}}{25}\right) + C$$$A