Integralen av $$$4 x \operatorname{atan}{\left(x \right)}$$$

Bestäm $$$\int 4 x \operatorname{atan}{\left(x \right)}\, dx$$$.

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=4$$$ och $$$f{\left(x \right)} = x \operatorname{atan}{\left(x \right)}$$$:

$${\color{red}{\int{4 x \operatorname{atan}{\left(x \right)} d x}}} = {\color{red}{\left(4 \int{x \operatorname{atan}{\left(x \right)} d x}\right)}}$$

För integralen $$$\int{x \operatorname{atan}{\left(x \right)} d x}$$$, 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 $$$\operatorname{dv}=x dx$$$.

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

Alltså,

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

Förenkla integranden:

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

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=\frac{1}{2}$$$ och $$$f{\left(x \right)} = \frac{x^{2}}{x^{2} + 1}$$$:

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

Skriv om och dela upp bråket:

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

Integrera termvis:

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

Tillämpa konstantregeln $$$\int c\, dx = c x$$$ med $$$c=1$$$:

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

Integralen av $$$\frac{1}{x^{2} + 1}$$$ är $$$\int{\frac{1}{x^{2} + 1} d x} = \operatorname{atan}{\left(x \right)}$$$:

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

Alltså,

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

Förenkla:

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

Lägg till integrationskonstanten:

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

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