Funktion $$$x \operatorname{atan}{\left(x \right)}$$$ integraali

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

Integraalin $$$\int{x \operatorname{atan}{\left(x \right)} d x}$$$ kohdalla käytä osittaisintegrointia $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

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

Tällöin $$$\operatorname{du}=\left(\operatorname{atan}{\left(x \right)}\right)^{\prime }dx=\frac{dx}{x^{2} + 1}$$$ (vaiheet ovat nähtävissä ») ja $$$\operatorname{v}=\int{x d x}=\frac{x^{2}}{2}$$$ (vaiheet ovat nähtävissä »).

Näin ollen,

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

Yksinkertaista integroitavaa:

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

Sovella vakiokertoimen sääntöä $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ käyttäen $$$c=\frac{1}{2}$$$ ja $$$f{\left(x \right)} = \frac{x^{2}}{x^{2} + 1}$$$:

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

Kirjoita murtolauseke uudelleen ja jaa se osamurtoihin:

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

Integroi termi kerrallaan:

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

Sovella vakiosääntöä $$$\int c\, dx = c x$$$ käyttäen $$$c=1$$$:

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

Funktion $$$\frac{1}{x^{2} + 1}$$$ integraali on $$$\int{\frac{1}{x^{2} + 1} d x} = \operatorname{atan}{\left(x \right)}$$$:

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

Näin ollen,

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

Sievennä:

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

Lisää integrointivakio:

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

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