Integral dari $$$\frac{2 \operatorname{atan}{\left(x \right)}}{x^{2}}$$$

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

Terapkan aturan pengali konstanta $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ dengan $$$c=2$$$ dan $$$f{\left(x \right)} = \frac{\operatorname{atan}{\left(x \right)}}{x^{2}}$$$:

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

Misalkan $$$u=\frac{1}{x}$$$.

Kemudian $$$du=\left(\frac{1}{x}\right)^{\prime }dx = - \frac{1}{x^{2}} dx$$$ (langkah-langkah dapat dilihat di »), dan kita memperoleh $$$\frac{dx}{x^{2}} = - du$$$.

Integral tersebut dapat ditulis ulang sebagai

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

Terapkan aturan pengali konstanta $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ dengan $$$c=-1$$$ dan $$$f{\left(u \right)} = \operatorname{acot}{\left(u \right)}$$$:

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

Untuk integral $$$\int{\operatorname{acot}{\left(u \right)} d u}$$$, gunakan integrasi parsial $$$\int \operatorname{t} \operatorname{dv} = \operatorname{t}\operatorname{v} - \int \operatorname{v} \operatorname{dt}$$$.

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

Maka $$$\operatorname{dt}=\left(\operatorname{acot}{\left(u \right)}\right)^{\prime }du=- \frac{1}{u^{2} + 1} du$$$ (langkah-langkah dapat dilihat di ») dan $$$\operatorname{v}=\int{1 d u}=u$$$ (langkah-langkah dapat dilihat di »).

Dengan demikian,

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

Misalkan $$$v=u^{2} + 1$$$.

Kemudian $$$dv=\left(u^{2} + 1\right)^{\prime }du = 2 u du$$$ (langkah-langkah dapat dilihat di »), dan kita memperoleh $$$u du = \frac{dv}{2}$$$.

Integral tersebut dapat ditulis ulang sebagai

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

Terapkan aturan pengali konstanta $$$\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv$$$ dengan $$$c=- \frac{1}{2}$$$ dan $$$f{\left(v \right)} = \frac{1}{v}$$$:

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

Integral dari $$$\frac{1}{v}$$$ adalah $$$\int{\frac{1}{v} d v} = \ln{\left(\left|{v}\right| \right)}$$$:

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

Ingat bahwa $$$v=u^{2} + 1$$$:

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

Ingat bahwa $$$u=\frac{1}{x}$$$:

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

Oleh karena itu,

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

Sederhanakan:

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

Tambahkan konstanta integrasi:

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

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