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

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

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

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

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

Alltså,

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

Låt $$$u=x^{2} + 1$$$ vara.

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

Integralen kan omskrivas som

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

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

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

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

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

Kom ihåg att $$$u=x^{2} + 1$$$:

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

Alltså,

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

Lägg till integrationskonstanten:

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

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