Integralen av $$$\frac{s^{2} x \sin{\left(1 \right)} \cos{\left(2 x \right)}}{c_{0}}$$$ med avseende på $$$x$$$

Bestäm $$$\int \frac{s^{2} x \sin{\left(1 \right)} \cos{\left(2 x \right)}}{c_{0}}\, dx$$$.

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=\frac{s^{2} \sin{\left(1 \right)}}{c_{0}}$$$ och $$$f{\left(x \right)} = x \cos{\left(2 x \right)}$$$:

$${\color{red}{\int{\frac{s^{2} x \sin{\left(1 \right)} \cos{\left(2 x \right)}}{c_{0}} d x}}} = {\color{red}{\frac{s^{2} \sin{\left(1 \right)} \int{x \cos{\left(2 x \right)} d x}}{c_{0}}}}$$

För integralen $$$\int{x \cos{\left(2 x \right)} d x}$$$, använd partiell integration $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Låt $$$\operatorname{u}=x$$$ och $$$\operatorname{dv}=\cos{\left(2 x \right)} dx$$$.

Då gäller $$$\operatorname{du}=\left(x\right)^{\prime }dx=1 dx$$$ (stegen kan ses ») och $$$\operatorname{v}=\int{\cos{\left(2 x \right)} d x}=\frac{\sin{\left(2 x \right)}}{2}$$$ (stegen kan ses »).

Alltså,

$$\frac{s^{2} \sin{\left(1 \right)} {\color{red}{\int{x \cos{\left(2 x \right)} d x}}}}{c_{0}}=\frac{s^{2} \sin{\left(1 \right)} {\color{red}{\left(x \cdot \frac{\sin{\left(2 x \right)}}{2}-\int{\frac{\sin{\left(2 x \right)}}{2} \cdot 1 d x}\right)}}}{c_{0}}=\frac{s^{2} \sin{\left(1 \right)} {\color{red}{\left(\frac{x \sin{\left(2 x \right)}}{2} - \int{\frac{\sin{\left(2 x \right)}}{2} d x}\right)}}}{c_{0}}$$

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

$$\frac{s^{2} \sin{\left(1 \right)} \left(\frac{x \sin{\left(2 x \right)}}{2} - {\color{red}{\int{\frac{\sin{\left(2 x \right)}}{2} d x}}}\right)}{c_{0}} = \frac{s^{2} \sin{\left(1 \right)} \left(\frac{x \sin{\left(2 x \right)}}{2} - {\color{red}{\left(\frac{\int{\sin{\left(2 x \right)} d x}}{2}\right)}}\right)}{c_{0}}$$

Låt $$$u=2 x$$$ vara.

Då $$$du=\left(2 x\right)^{\prime }dx = 2 dx$$$ (stegen kan ses »), och vi har att $$$dx = \frac{du}{2}$$$.

Alltså,

$$\frac{s^{2} \sin{\left(1 \right)} \left(\frac{x \sin{\left(2 x \right)}}{2} - \frac{{\color{red}{\int{\sin{\left(2 x \right)} d x}}}}{2}\right)}{c_{0}} = \frac{s^{2} \sin{\left(1 \right)} \left(\frac{x \sin{\left(2 x \right)}}{2} - \frac{{\color{red}{\int{\frac{\sin{\left(u \right)}}{2} d u}}}}{2}\right)}{c_{0}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ med $$$c=\frac{1}{2}$$$ och $$$f{\left(u \right)} = \sin{\left(u \right)}$$$:

$$\frac{s^{2} \sin{\left(1 \right)} \left(\frac{x \sin{\left(2 x \right)}}{2} - \frac{{\color{red}{\int{\frac{\sin{\left(u \right)}}{2} d u}}}}{2}\right)}{c_{0}} = \frac{s^{2} \sin{\left(1 \right)} \left(\frac{x \sin{\left(2 x \right)}}{2} - \frac{{\color{red}{\left(\frac{\int{\sin{\left(u \right)} d u}}{2}\right)}}}{2}\right)}{c_{0}}$$

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

$$\frac{s^{2} \sin{\left(1 \right)} \left(\frac{x \sin{\left(2 x \right)}}{2} - \frac{{\color{red}{\int{\sin{\left(u \right)} d u}}}}{4}\right)}{c_{0}} = \frac{s^{2} \sin{\left(1 \right)} \left(\frac{x \sin{\left(2 x \right)}}{2} - \frac{{\color{red}{\left(- \cos{\left(u \right)}\right)}}}{4}\right)}{c_{0}}$$

Kom ihåg att $$$u=2 x$$$:

$$\frac{s^{2} \sin{\left(1 \right)} \left(\frac{x \sin{\left(2 x \right)}}{2} + \frac{\cos{\left({\color{red}{u}} \right)}}{4}\right)}{c_{0}} = \frac{s^{2} \sin{\left(1 \right)} \left(\frac{x \sin{\left(2 x \right)}}{2} + \frac{\cos{\left({\color{red}{\left(2 x\right)}} \right)}}{4}\right)}{c_{0}}$$

Alltså,

$$\int{\frac{s^{2} x \sin{\left(1 \right)} \cos{\left(2 x \right)}}{c_{0}} d x} = \frac{s^{2} \left(\frac{x \sin{\left(2 x \right)}}{2} + \frac{\cos{\left(2 x \right)}}{4}\right) \sin{\left(1 \right)}}{c_{0}}$$

Förenkla:

$$\int{\frac{s^{2} x \sin{\left(1 \right)} \cos{\left(2 x \right)}}{c_{0}} d x} = \frac{s^{2} \left(2 x \sin{\left(2 x \right)} + \cos{\left(2 x \right)}\right) \sin{\left(1 \right)}}{4 c_{0}}$$

Lägg till integrationskonstanten:

$$\int{\frac{s^{2} x \sin{\left(1 \right)} \cos{\left(2 x \right)}}{c_{0}} d x} = \frac{s^{2} \left(2 x \sin{\left(2 x \right)} + \cos{\left(2 x \right)}\right) \sin{\left(1 \right)}}{4 c_{0}}+C$$

$$$\int \frac{s^{2} x \sin{\left(1 \right)} \cos{\left(2 x \right)}}{c_{0}}\, dx = \frac{s^{2} \left(2 x \sin{\left(2 x \right)} + \cos{\left(2 x \right)}\right) \sin{\left(1 \right)}}{4 c_{0}} + C$$$A