Integralen av $$$\operatorname{atan}{\left(\frac{x}{2} \right)}$$$

Bestäm $$$\int \operatorname{atan}{\left(\frac{x}{2} \right)}\, dx$$$.

Låt $$$u=\frac{x}{2}$$$ vara.

Då $$$du=\left(\frac{x}{2}\right)^{\prime }dx = \frac{dx}{2}$$$ (stegen kan ses »), och vi har att $$$dx = 2 du$$$.

Integralen blir

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

Tillämpa konstantfaktorregeln $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ med $$$c=2$$$ och $$$f{\left(u \right)} = \operatorname{atan}{\left(u \right)}$$$:

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

För integralen $$$\int{\operatorname{atan}{\left(u \right)} d u}$$$, använd partiell integration $$$\int \operatorname{\kappa} \operatorname{dv} = \operatorname{\kappa}\operatorname{v} - \int \operatorname{v} \operatorname{d\kappa}$$$.

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

Då gäller $$$\operatorname{d\kappa}=\left(\operatorname{atan}{\left(u \right)}\right)^{\prime }du=\frac{du}{u^{2} + 1}$$$ (stegen kan ses ») och $$$\operatorname{v}=\int{1 d u}=u$$$ (stegen kan ses »).

Alltså,

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

Låt $$$v=u^{2} + 1$$$ vara.

Då $$$dv=\left(u^{2} + 1\right)^{\prime }du = 2 u du$$$ (stegen kan ses »), och vi har att $$$u du = \frac{dv}{2}$$$.

Alltså,

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

Tillämpa konstantfaktorregeln $$$\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv$$$ med $$$c=\frac{1}{2}$$$ och $$$f{\left(v \right)} = \frac{1}{v}$$$:

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

Integralen av $$$\frac{1}{v}$$$ är $$$\int{\frac{1}{v} d v} = \ln{\left(\left|{v}\right| \right)}$$$:

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

Kom ihåg att $$$v=u^{2} + 1$$$:

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

Kom ihåg att $$$u=\frac{x}{2}$$$:

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

Alltså,

$$\int{\operatorname{atan}{\left(\frac{x}{2} \right)} d x} = x \operatorname{atan}{\left(\frac{x}{2} \right)} - \ln{\left(\frac{x^{2}}{4} + 1 \right)}$$

Lägg till integrationskonstanten:

$$\int{\operatorname{atan}{\left(\frac{x}{2} \right)} d x} = x \operatorname{atan}{\left(\frac{x}{2} \right)} - \ln{\left(\frac{x^{2}}{4} + 1 \right)}+C$$

$$$\int \operatorname{atan}{\left(\frac{x}{2} \right)}\, dx = \left(x \operatorname{atan}{\left(\frac{x}{2} \right)} - \ln\left(\frac{x^{2}}{4} + 1\right)\right) + C$$$A