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

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

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

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

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

Alltså,

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

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

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

Använd potensreduceringsformeln $$$\sin^{2}{\left(\alpha \right)} = \frac{1}{2} - \frac{\cos{\left(2 \alpha \right)}}{2}$$$ med $$$\alpha=x$$$:

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

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

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

Integrera termvis:

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

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

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

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,

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

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

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

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

$$\frac{x \sin^{2}{\left(x \right)}}{2} - \frac{x}{4} + \frac{{\color{red}{\int{\cos{\left(u \right)} d u}}}}{8} = \frac{x \sin^{2}{\left(x \right)}}{2} - \frac{x}{4} + \frac{{\color{red}{\sin{\left(u \right)}}}}{8}$$

Kom ihåg att $$$u=2 x$$$:

$$\frac{x \sin^{2}{\left(x \right)}}{2} - \frac{x}{4} + \frac{\sin{\left({\color{red}{u}} \right)}}{8} = \frac{x \sin^{2}{\left(x \right)}}{2} - \frac{x}{4} + \frac{\sin{\left({\color{red}{\left(2 x\right)}} \right)}}{8}$$

Alltså,

$$\int{x \sin{\left(x \right)} \cos{\left(x \right)} d x} = \frac{x \sin^{2}{\left(x \right)}}{2} - \frac{x}{4} + \frac{\sin{\left(2 x \right)}}{8}$$

Lägg till integrationskonstanten:

$$\int{x \sin{\left(x \right)} \cos{\left(x \right)} d x} = \frac{x \sin^{2}{\left(x \right)}}{2} - \frac{x}{4} + \frac{\sin{\left(2 x \right)}}{8}+C$$

$$$\int x \sin{\left(x \right)} \cos{\left(x \right)}\, dx = \left(\frac{x \sin^{2}{\left(x \right)}}{2} - \frac{x}{4} + \frac{\sin{\left(2 x \right)}}{8}\right) + C$$$A