Integraal van $$$\frac{1}{\sqrt{x} \csc{\left(\sqrt{x} \right)}}$$$

Bepaal $$$\int \frac{1}{\sqrt{x} \csc{\left(\sqrt{x} \right)}}\, dx$$$.

Zij $$$u=\sqrt{x}$$$.

Dan $$$du=\left(\sqrt{x}\right)^{\prime }dx = \frac{1}{2 \sqrt{x}} dx$$$ (de stappen zijn te zien »), en dan geldt dat $$$\frac{dx}{\sqrt{x}} = 2 du$$$.

De integraal wordt

$${\color{red}{\int{\frac{1}{\sqrt{x} \csc{\left(\sqrt{x} \right)}} d x}}} = {\color{red}{\int{\frac{2}{\csc{\left(u \right)}} d u}}}$$

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

$${\color{red}{\int{\frac{2}{\csc{\left(u \right)}} d u}}} = {\color{red}{\left(2 \int{\frac{1}{\csc{\left(u \right)}} d u}\right)}}$$

Herschrijf de integrand in termen van de sinus:

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

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

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

We herinneren eraan dat $$$u=\sqrt{x}$$$:

$$- 2 \cos{\left({\color{red}{u}} \right)} = - 2 \cos{\left({\color{red}{\sqrt{x}}} \right)}$$

Dus,

$$\int{\frac{1}{\sqrt{x} \csc{\left(\sqrt{x} \right)}} d x} = - 2 \cos{\left(\sqrt{x} \right)}$$

Voeg de integratieconstante toe:

$$\int{\frac{1}{\sqrt{x} \csc{\left(\sqrt{x} \right)}} d x} = - 2 \cos{\left(\sqrt{x} \right)}+C$$

$$$\int \frac{1}{\sqrt{x} \csc{\left(\sqrt{x} \right)}}\, dx = - 2 \cos{\left(\sqrt{x} \right)} + C$$$A