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

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

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=2$$$ och $$$f{\left(x \right)} = \frac{\operatorname{atan}{\left(x \right)}}{x^{2}}$$$:

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

Låt $$$u=\frac{1}{x}$$$ vara.

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

Alltså,

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

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

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

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

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

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

Alltså,

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

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,

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

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

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

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

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

Kom ihåg att $$$v=u^{2} + 1$$$:

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

Kom ihåg att $$$u=\frac{1}{x}$$$:

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

Alltså,

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

Förenkla:

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

Lägg till integrationskonstanten:

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

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