Integraal van $$$\cos{\left(\theta \right)} \cot{\left(\theta \right)}$$$

Bepaal $$$\int \cos{\left(\theta \right)} \cot{\left(\theta \right)}\, d\theta$$$.

Herschrijf de integraand:

$${\color{red}{\int{\cos{\left(\theta \right)} \cot{\left(\theta \right)} d \theta}}} = {\color{red}{\int{\frac{\cos^{2}{\left(\theta \right)}}{\sin{\left(\theta \right)}} d \theta}}}$$

Vermenigvuldig de teller en de noemer met een sinus en schrijf de rest in termen van de cosinus, met behulp van de formule $$$\sin^2\left(\alpha \right)=-\cos^2\left(\alpha \right)+1$$$ met $$$\alpha=\theta$$$:

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

Zij $$$u=\cos{\left(\theta \right)}$$$.

Dan $$$du=\left(\cos{\left(\theta \right)}\right)^{\prime }d\theta = - \sin{\left(\theta \right)} d\theta$$$ (de stappen zijn te zien »), en dan geldt dat $$$\sin{\left(\theta \right)} d\theta = - du$$$.

Dus,

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

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

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

Aangezien de graad van de teller niet kleiner is dan die van de noemer, voer een staartdeling van polynomen uit (stappen zijn te zien »):

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

Integreer termgewijs:

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

Pas de constantenregel $$$\int c\, du = c u$$$ toe met $$$c=1$$$:

$$- \int{\frac{1}{1 - u^{2}} d u} + {\color{red}{\int{1 d u}}} = - \int{\frac{1}{1 - u^{2}} d u} + {\color{red}{u}}$$

Voer een ontbinding in partiële breuken uit (stappen zijn te zien »):

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

Integreer termgewijs:

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

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 $$$f{\left(u \right)} = \frac{1}{u + 1}$$$:

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

Zij $$$v=u + 1$$$.

Dan $$$dv=\left(u + 1\right)^{\prime }du = 1 du$$$ (de stappen zijn te zien »), en dan geldt dat $$$du = dv$$$.

Dus,

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

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

$$u + \int{\frac{1}{2 \left(u - 1\right)} d u} - \frac{{\color{red}{\int{\frac{1}{v} d v}}}}{2} = u + \int{\frac{1}{2 \left(u - 1\right)} d u} - \frac{{\color{red}{\ln{\left(\left|{v}\right| \right)}}}}{2}$$

We herinneren eraan dat $$$v=u + 1$$$:

$$u - \frac{\ln{\left(\left|{{\color{red}{v}}}\right| \right)}}{2} + \int{\frac{1}{2 \left(u - 1\right)} d u} = u - \frac{\ln{\left(\left|{{\color{red}{\left(u + 1\right)}}}\right| \right)}}{2} + \int{\frac{1}{2 \left(u - 1\right)} d u}$$

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 $$$f{\left(u \right)} = \frac{1}{u - 1}$$$:

$$u - \frac{\ln{\left(\left|{u + 1}\right| \right)}}{2} + {\color{red}{\int{\frac{1}{2 \left(u - 1\right)} d u}}} = u - \frac{\ln{\left(\left|{u + 1}\right| \right)}}{2} + {\color{red}{\left(\frac{\int{\frac{1}{u - 1} d u}}{2}\right)}}$$

Zij $$$v=u - 1$$$.

Dan $$$dv=\left(u - 1\right)^{\prime }du = 1 du$$$ (de stappen zijn te zien »), en dan geldt dat $$$du = dv$$$.

Dus,

$$u - \frac{\ln{\left(\left|{u + 1}\right| \right)}}{2} + \frac{{\color{red}{\int{\frac{1}{u - 1} d u}}}}{2} = u - \frac{\ln{\left(\left|{u + 1}\right| \right)}}{2} + \frac{{\color{red}{\int{\frac{1}{v} d v}}}}{2}$$

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

$$u - \frac{\ln{\left(\left|{u + 1}\right| \right)}}{2} + \frac{{\color{red}{\int{\frac{1}{v} d v}}}}{2} = u - \frac{\ln{\left(\left|{u + 1}\right| \right)}}{2} + \frac{{\color{red}{\ln{\left(\left|{v}\right| \right)}}}}{2}$$

We herinneren eraan dat $$$v=u - 1$$$:

$$u - \frac{\ln{\left(\left|{u + 1}\right| \right)}}{2} + \frac{\ln{\left(\left|{{\color{red}{v}}}\right| \right)}}{2} = u - \frac{\ln{\left(\left|{u + 1}\right| \right)}}{2} + \frac{\ln{\left(\left|{{\color{red}{\left(u - 1\right)}}}\right| \right)}}{2}$$

We herinneren eraan dat $$$u=\cos{\left(\theta \right)}$$$:

$$\frac{\ln{\left(\left|{-1 + {\color{red}{u}}}\right| \right)}}{2} - \frac{\ln{\left(\left|{1 + {\color{red}{u}}}\right| \right)}}{2} + {\color{red}{u}} = \frac{\ln{\left(\left|{-1 + {\color{red}{\cos{\left(\theta \right)}}}}\right| \right)}}{2} - \frac{\ln{\left(\left|{1 + {\color{red}{\cos{\left(\theta \right)}}}}\right| \right)}}{2} + {\color{red}{\cos{\left(\theta \right)}}}$$

Dus,

$$\int{\cos{\left(\theta \right)} \cot{\left(\theta \right)} d \theta} = \frac{\ln{\left(\left|{\cos{\left(\theta \right)} - 1}\right| \right)}}{2} - \frac{\ln{\left(\left|{\cos{\left(\theta \right)} + 1}\right| \right)}}{2} + \cos{\left(\theta \right)}$$

Voeg de integratieconstante toe:

$$\int{\cos{\left(\theta \right)} \cot{\left(\theta \right)} d \theta} = \frac{\ln{\left(\left|{\cos{\left(\theta \right)} - 1}\right| \right)}}{2} - \frac{\ln{\left(\left|{\cos{\left(\theta \right)} + 1}\right| \right)}}{2} + \cos{\left(\theta \right)}+C$$

$$$\int \cos{\left(\theta \right)} \cot{\left(\theta \right)}\, d\theta = \left(\frac{\ln\left(\left|{\cos{\left(\theta \right)} - 1}\right|\right)}{2} - \frac{\ln\left(\left|{\cos{\left(\theta \right)} + 1}\right|\right)}{2} + \cos{\left(\theta \right)}\right) + C$$$A