Integraal van $$$- \csc{\left(x \right)}$$$

Bepaal $$$\int \left(- \csc{\left(x \right)}\right)\, dx$$$.

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

$${\color{red}{\int{\left(- \csc{\left(x \right)}\right)d x}}} = {\color{red}{\left(- \int{\csc{\left(x \right)} d x}\right)}}$$

Herschrijf de cosecans als $$$\csc\left(x\right)=\frac{1}{\sin\left(x\right)}$$$:

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

Herschrijf de sinus met behulp van de formule voor de dubbele hoek $$$\sin\left(x\right)=2\sin\left(\frac{x}{2}\right)\cos\left(\frac{x}{2}\right)$$$:

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

Vermenigvuldig de teller en de noemer met $$$\sec^2\left(\frac{x}{2} \right)$$$:

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

Zij $$$u=\tan{\left(\frac{x}{2} \right)}$$$.

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

Dus,

$$- {\color{red}{\int{\frac{\sec^{2}{\left(\frac{x}{2} \right)}}{2 \tan{\left(\frac{x}{2} \right)}} d x}}} = - {\color{red}{\int{\frac{1}{u} d u}}}$$

De integraal van $$$\frac{1}{u}$$$ is $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$:

$$- {\color{red}{\int{\frac{1}{u} d u}}} = - {\color{red}{\ln{\left(\left|{u}\right| \right)}}}$$

We herinneren eraan dat $$$u=\tan{\left(\frac{x}{2} \right)}$$$:

$$- \ln{\left(\left|{{\color{red}{u}}}\right| \right)} = - \ln{\left(\left|{{\color{red}{\tan{\left(\frac{x}{2} \right)}}}}\right| \right)}$$

Dus,

$$\int{\left(- \csc{\left(x \right)}\right)d x} = - \ln{\left(\left|{\tan{\left(\frac{x}{2} \right)}}\right| \right)}$$

Voeg de integratieconstante toe:

$$\int{\left(- \csc{\left(x \right)}\right)d x} = - \ln{\left(\left|{\tan{\left(\frac{x}{2} \right)}}\right| \right)}+C$$

$$$\int \left(- \csc{\left(x \right)}\right)\, dx = - \ln\left(\left|{\tan{\left(\frac{x}{2} \right)}}\right|\right) + C$$$A