Integraal van $$$\frac{\operatorname{atan}{\left(\sqrt{x} \right)}}{\sqrt{x}}$$$

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

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 $$$\frac{dx}{\sqrt{x}} = 2 du$$$.

Dus,

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

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

$${\color{red}{\int{2 \operatorname{atan}{\left(u \right)} d u}}} = {\color{red}{\left(2 \int{\operatorname{atan}{\left(u \right)} d u}\right)}}$$

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

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

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

De integraal kan worden herschreven als

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

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

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

Dus,

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

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

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

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

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

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

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

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

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

Dus,

$$\int{\frac{\operatorname{atan}{\left(\sqrt{x} \right)}}{\sqrt{x}} d x} = 2 \sqrt{x} \operatorname{atan}{\left(\sqrt{x} \right)} - \ln{\left(x + 1 \right)}$$

Voeg de integratieconstante toe:

$$\int{\frac{\operatorname{atan}{\left(\sqrt{x} \right)}}{\sqrt{x}} d x} = 2 \sqrt{x} \operatorname{atan}{\left(\sqrt{x} \right)} - \ln{\left(x + 1 \right)}+C$$

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