Integraal van $$$x^{3} \sin{\left(x \right)}$$$

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

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

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

Dan $$$\operatorname{du}=\left(x^{3}\right)^{\prime }dx=3 x^{2} dx$$$ (de stappen zijn te zien ») en $$$\operatorname{v}=\int{\sin{\left(x \right)} d x}=- \cos{\left(x \right)}$$$ (de stappen zijn te zien »).

Dus,

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

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

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

Voor de integraal $$$\int{x^{2} \cos{\left(x \right)} d x}$$$, gebruik partiële integratie $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Zij $$$\operatorname{u}=x^{2}$$$ en $$$\operatorname{dv}=\cos{\left(x \right)} dx$$$.

Dan $$$\operatorname{du}=\left(x^{2}\right)^{\prime }dx=2 x dx$$$ (de stappen zijn te zien ») en $$$\operatorname{v}=\int{\cos{\left(x \right)} d x}=\sin{\left(x \right)}$$$ (de stappen zijn te zien »).

De integraal kan worden herschreven als

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

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

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

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

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

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

Dus,

$$- x^{3} \cos{\left(x \right)} + 3 x^{2} \sin{\left(x \right)} - 6 {\color{red}{\int{x \sin{\left(x \right)} d x}}}=- x^{3} \cos{\left(x \right)} + 3 x^{2} \sin{\left(x \right)} - 6 {\color{red}{\left(x \cdot \left(- \cos{\left(x \right)}\right)-\int{\left(- \cos{\left(x \right)}\right) \cdot 1 d x}\right)}}=- x^{3} \cos{\left(x \right)} + 3 x^{2} \sin{\left(x \right)} - 6 {\color{red}{\left(- x \cos{\left(x \right)} - \int{\left(- \cos{\left(x \right)}\right)d x}\right)}}$$

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

$$- x^{3} \cos{\left(x \right)} + 3 x^{2} \sin{\left(x \right)} + 6 x \cos{\left(x \right)} + 6 {\color{red}{\int{\left(- \cos{\left(x \right)}\right)d x}}} = - x^{3} \cos{\left(x \right)} + 3 x^{2} \sin{\left(x \right)} + 6 x \cos{\left(x \right)} + 6 {\color{red}{\left(- \int{\cos{\left(x \right)} d x}\right)}}$$

De integraal van de cosinus is $$$\int{\cos{\left(x \right)} d x} = \sin{\left(x \right)}$$$:

$$- x^{3} \cos{\left(x \right)} + 3 x^{2} \sin{\left(x \right)} + 6 x \cos{\left(x \right)} - 6 {\color{red}{\int{\cos{\left(x \right)} d x}}} = - x^{3} \cos{\left(x \right)} + 3 x^{2} \sin{\left(x \right)} + 6 x \cos{\left(x \right)} - 6 {\color{red}{\sin{\left(x \right)}}}$$

Dus,

$$\int{x^{3} \sin{\left(x \right)} d x} = - x^{3} \cos{\left(x \right)} + 3 x^{2} \sin{\left(x \right)} + 6 x \cos{\left(x \right)} - 6 \sin{\left(x \right)}$$

Voeg de integratieconstante toe:

$$\int{x^{3} \sin{\left(x \right)} d x} = - x^{3} \cos{\left(x \right)} + 3 x^{2} \sin{\left(x \right)} + 6 x \cos{\left(x \right)} - 6 \sin{\left(x \right)}+C$$

$$$\int x^{3} \sin{\left(x \right)}\, dx = \left(- x^{3} \cos{\left(x \right)} + 3 x^{2} \sin{\left(x \right)} + 6 x \cos{\left(x \right)} - 6 \sin{\left(x \right)}\right) + C$$$A