Integraal van $$$\operatorname{acos}{\left(3 x \right)}$$$

Bepaal $$$\int \operatorname{acos}{\left(3 x \right)}\, dx$$$.

Zij $$$u=3 x$$$.

Dan $$$du=\left(3 x\right)^{\prime }dx = 3 dx$$$ (de stappen zijn te zien »), en dan geldt dat $$$dx = \frac{du}{3}$$$.

Dus,

$${\color{red}{\int{\operatorname{acos}{\left(3 x \right)} d x}}} = {\color{red}{\int{\frac{\operatorname{acos}{\left(u \right)}}{3} d u}}}$$

Pas de constante-veelvoudregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ toe met $$$c=\frac{1}{3}$$$ en $$$f{\left(u \right)} = \operatorname{acos}{\left(u \right)}$$$:

$${\color{red}{\int{\frac{\operatorname{acos}{\left(u \right)}}{3} d u}}} = {\color{red}{\left(\frac{\int{\operatorname{acos}{\left(u \right)} d u}}{3}\right)}}$$

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

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

Dan $$$\operatorname{dm}=\left(\operatorname{acos}{\left(u \right)}\right)^{\prime }du=- \frac{1}{\sqrt{1 - u^{2}}} du$$$ (de stappen zijn te zien ») en $$$\operatorname{v}=\int{1 d u}=u$$$ (de stappen zijn te zien »).

Dus,

$$\frac{{\color{red}{\int{\operatorname{acos}{\left(u \right)} d u}}}}{3}=\frac{{\color{red}{\left(\operatorname{acos}{\left(u \right)} \cdot u-\int{u \cdot \left(- \frac{1}{\sqrt{1 - u^{2}}}\right) d u}\right)}}}{3}=\frac{{\color{red}{\left(u \operatorname{acos}{\left(u \right)} - \int{\left(- \frac{u}{\sqrt{1 - u^{2}}}\right)d u}\right)}}}{3}$$

Zij $$$v=1 - u^{2}$$$.

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

De integraal wordt

$$\frac{u \operatorname{acos}{\left(u \right)}}{3} - \frac{{\color{red}{\int{\left(- \frac{u}{\sqrt{1 - u^{2}}}\right)d u}}}}{3} = \frac{u \operatorname{acos}{\left(u \right)}}{3} - \frac{{\color{red}{\int{\frac{1}{2 \sqrt{v}} d v}}}}{3}$$

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

$$\frac{u \operatorname{acos}{\left(u \right)}}{3} - \frac{{\color{red}{\int{\frac{1}{2 \sqrt{v}} d v}}}}{3} = \frac{u \operatorname{acos}{\left(u \right)}}{3} - \frac{{\color{red}{\left(\frac{\int{\frac{1}{\sqrt{v}} d v}}{2}\right)}}}{3}$$

Pas de machtsregel $$$\int v^{n}\, dv = \frac{v^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ toe met $$$n=- \frac{1}{2}$$$:

$$\frac{u \operatorname{acos}{\left(u \right)}}{3} - \frac{{\color{red}{\int{\frac{1}{\sqrt{v}} d v}}}}{6}=\frac{u \operatorname{acos}{\left(u \right)}}{3} - \frac{{\color{red}{\int{v^{- \frac{1}{2}} d v}}}}{6}=\frac{u \operatorname{acos}{\left(u \right)}}{3} - \frac{{\color{red}{\frac{v^{- \frac{1}{2} + 1}}{- \frac{1}{2} + 1}}}}{6}=\frac{u \operatorname{acos}{\left(u \right)}}{3} - \frac{{\color{red}{\left(2 v^{\frac{1}{2}}\right)}}}{6}=\frac{u \operatorname{acos}{\left(u \right)}}{3} - \frac{{\color{red}{\left(2 \sqrt{v}\right)}}}{6}$$

We herinneren eraan dat $$$v=1 - u^{2}$$$:

$$\frac{u \operatorname{acos}{\left(u \right)}}{3} - \frac{\sqrt{{\color{red}{v}}}}{3} = \frac{u \operatorname{acos}{\left(u \right)}}{3} - \frac{\sqrt{{\color{red}{\left(1 - u^{2}\right)}}}}{3}$$

We herinneren eraan dat $$$u=3 x$$$:

$$- \frac{\sqrt{1 - {\color{red}{u}}^{2}}}{3} + \frac{{\color{red}{u}} \operatorname{acos}{\left({\color{red}{u}} \right)}}{3} = - \frac{\sqrt{1 - {\color{red}{\left(3 x\right)}}^{2}}}{3} + \frac{{\color{red}{\left(3 x\right)}} \operatorname{acos}{\left({\color{red}{\left(3 x\right)}} \right)}}{3}$$

Dus,

$$\int{\operatorname{acos}{\left(3 x \right)} d x} = x \operatorname{acos}{\left(3 x \right)} - \frac{\sqrt{1 - 9 x^{2}}}{3}$$

Voeg de integratieconstante toe:

$$\int{\operatorname{acos}{\left(3 x \right)} d x} = x \operatorname{acos}{\left(3 x \right)} - \frac{\sqrt{1 - 9 x^{2}}}{3}+C$$

$$$\int \operatorname{acos}{\left(3 x \right)}\, dx = \left(x \operatorname{acos}{\left(3 x \right)} - \frac{\sqrt{1 - 9 x^{2}}}{3}\right) + C$$$A