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

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

Zij $$$u=x^{2}$$$.

Dan $$$du=\left(x^{2}\right)^{\prime }dx = 2 x dx$$$ (de stappen zijn te zien »), en dan geldt dat $$$x dx = \frac{du}{2}$$$.

Dus,

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

Pas de constante-veelvoudregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ toe met $$$c=\frac{1}{2}$$$ en $$$f{\left(u \right)} = u \cos{\left(u \right)}$$$:

$${\color{red}{\int{\frac{u \cos{\left(u \right)}}{2} d u}}} = {\color{red}{\left(\frac{\int{u \cos{\left(u \right)} d u}}{2}\right)}}$$

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

Zij $$$\operatorname{m}=u$$$ en $$$\operatorname{dv}=\cos{\left(u \right)} du$$$.

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

De integraal wordt

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

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

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

We herinneren eraan dat $$$u=x^{2}$$$:

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

Dus,

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

Vereenvoudig:

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

Voeg de integratieconstante toe:

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

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