Integraal van $$$x e^{2} \cos{\left(3 x \right)}$$$

Bepaal $$$\int x e^{2} \cos{\left(3 x \right)}\, dx$$$.

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=e^{2}$$$ en $$$f{\left(x \right)} = x \cos{\left(3 x \right)}$$$:

$${\color{red}{\int{x e^{2} \cos{\left(3 x \right)} d x}}} = {\color{red}{e^{2} \int{x \cos{\left(3 x \right)} d x}}}$$

Voor de integraal $$$\int{x \cos{\left(3 x \right)} d x}$$$, gebruik partiële integratie $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Zij $$$\operatorname{u}=x$$$ en $$$\operatorname{dv}=\cos{\left(3 x \right)} dx$$$.

Dan $$$\operatorname{du}=\left(x\right)^{\prime }dx=1 dx$$$ (de stappen zijn te zien ») en $$$\operatorname{v}=\int{\cos{\left(3 x \right)} d x}=\frac{\sin{\left(3 x \right)}}{3}$$$ (de stappen zijn te zien »).

De integraal wordt

$$e^{2} {\color{red}{\int{x \cos{\left(3 x \right)} d x}}}=e^{2} {\color{red}{\left(x \cdot \frac{\sin{\left(3 x \right)}}{3}-\int{\frac{\sin{\left(3 x \right)}}{3} \cdot 1 d x}\right)}}=e^{2} {\color{red}{\left(\frac{x \sin{\left(3 x \right)}}{3} - \int{\frac{\sin{\left(3 x \right)}}{3} d x}\right)}}$$

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=\frac{1}{3}$$$ en $$$f{\left(x \right)} = \sin{\left(3 x \right)}$$$:

$$e^{2} \left(\frac{x \sin{\left(3 x \right)}}{3} - {\color{red}{\int{\frac{\sin{\left(3 x \right)}}{3} d x}}}\right) = e^{2} \left(\frac{x \sin{\left(3 x \right)}}{3} - {\color{red}{\left(\frac{\int{\sin{\left(3 x \right)} d x}}{3}\right)}}\right)$$

Zij $$$u=3 x$$$.

Dan $$$du=\left(3 x\right)^{\prime }dx = 3 dx$$$ (de stappen zijn te zien »), en dan geldt dat $$$dx = \frac{du}{3}$$$.

Dus,

$$e^{2} \left(\frac{x \sin{\left(3 x \right)}}{3} - \frac{{\color{red}{\int{\sin{\left(3 x \right)} d x}}}}{3}\right) = e^{2} \left(\frac{x \sin{\left(3 x \right)}}{3} - \frac{{\color{red}{\int{\frac{\sin{\left(u \right)}}{3} d u}}}}{3}\right)$$

Pas de constante-veelvoudregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ toe met $$$c=\frac{1}{3}$$$ en $$$f{\left(u \right)} = \sin{\left(u \right)}$$$:

$$e^{2} \left(\frac{x \sin{\left(3 x \right)}}{3} - \frac{{\color{red}{\int{\frac{\sin{\left(u \right)}}{3} d u}}}}{3}\right) = e^{2} \left(\frac{x \sin{\left(3 x \right)}}{3} - \frac{{\color{red}{\left(\frac{\int{\sin{\left(u \right)} d u}}{3}\right)}}}{3}\right)$$

De integraal van de sinus is $$$\int{\sin{\left(u \right)} d u} = - \cos{\left(u \right)}$$$:

$$e^{2} \left(\frac{x \sin{\left(3 x \right)}}{3} - \frac{{\color{red}{\int{\sin{\left(u \right)} d u}}}}{9}\right) = e^{2} \left(\frac{x \sin{\left(3 x \right)}}{3} - \frac{{\color{red}{\left(- \cos{\left(u \right)}\right)}}}{9}\right)$$

We herinneren eraan dat $$$u=3 x$$$:

$$e^{2} \left(\frac{x \sin{\left(3 x \right)}}{3} + \frac{\cos{\left({\color{red}{u}} \right)}}{9}\right) = e^{2} \left(\frac{x \sin{\left(3 x \right)}}{3} + \frac{\cos{\left({\color{red}{\left(3 x\right)}} \right)}}{9}\right)$$

Dus,

$$\int{x e^{2} \cos{\left(3 x \right)} d x} = \left(\frac{x \sin{\left(3 x \right)}}{3} + \frac{\cos{\left(3 x \right)}}{9}\right) e^{2}$$

Vereenvoudig:

$$\int{x e^{2} \cos{\left(3 x \right)} d x} = \frac{\left(3 x \sin{\left(3 x \right)} + \cos{\left(3 x \right)}\right) e^{2}}{9}$$

Voeg de integratieconstante toe:

$$\int{x e^{2} \cos{\left(3 x \right)} d x} = \frac{\left(3 x \sin{\left(3 x \right)} + \cos{\left(3 x \right)}\right) e^{2}}{9}+C$$

$$$\int x e^{2} \cos{\left(3 x \right)}\, dx = \frac{\left(3 x \sin{\left(3 x \right)} + \cos{\left(3 x \right)}\right) e^{2}}{9} + C$$$A