Integral dari $$$\operatorname{atan}{\left(7 t \right)}$$$

Temukan $$$\int \operatorname{atan}{\left(7 t \right)}\, dt$$$.

Misalkan $$$u=7 t$$$.

Kemudian $$$du=\left(7 t\right)^{\prime }dt = 7 dt$$$ (langkah-langkah dapat dilihat di »), dan kita memperoleh $$$dt = \frac{du}{7}$$$.

Jadi,

$${\color{red}{\int{\operatorname{atan}{\left(7 t \right)} d t}}} = {\color{red}{\int{\frac{\operatorname{atan}{\left(u \right)}}{7} d u}}}$$

Terapkan aturan pengali konstanta $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ dengan $$$c=\frac{1}{7}$$$ dan $$$f{\left(u \right)} = \operatorname{atan}{\left(u \right)}$$$:

$${\color{red}{\int{\frac{\operatorname{atan}{\left(u \right)}}{7} d u}}} = {\color{red}{\left(\frac{\int{\operatorname{atan}{\left(u \right)} d u}}{7}\right)}}$$

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

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

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

Dengan demikian,

$$\frac{{\color{red}{\int{\operatorname{atan}{\left(u \right)} d u}}}}{7}=\frac{{\color{red}{\left(\operatorname{atan}{\left(u \right)} \cdot u-\int{u \cdot \frac{1}{u^{2} + 1} d u}\right)}}}{7}=\frac{{\color{red}{\left(u \operatorname{atan}{\left(u \right)} - \int{\frac{u}{u^{2} + 1} d u}\right)}}}{7}$$

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 $$$u du = \frac{dv}{2}$$$.

Oleh karena itu,

$$\frac{u \operatorname{atan}{\left(u \right)}}{7} - \frac{{\color{red}{\int{\frac{u}{u^{2} + 1} d u}}}}{7} = \frac{u \operatorname{atan}{\left(u \right)}}{7} - \frac{{\color{red}{\int{\frac{1}{2 v} d v}}}}{7}$$

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

$$\frac{u \operatorname{atan}{\left(u \right)}}{7} - \frac{{\color{red}{\int{\frac{1}{2 v} d v}}}}{7} = \frac{u \operatorname{atan}{\left(u \right)}}{7} - \frac{{\color{red}{\left(\frac{\int{\frac{1}{v} d v}}{2}\right)}}}{7}$$

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

$$\frac{u \operatorname{atan}{\left(u \right)}}{7} - \frac{{\color{red}{\int{\frac{1}{v} d v}}}}{14} = \frac{u \operatorname{atan}{\left(u \right)}}{7} - \frac{{\color{red}{\ln{\left(\left|{v}\right| \right)}}}}{14}$$

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

$$\frac{u \operatorname{atan}{\left(u \right)}}{7} - \frac{\ln{\left(\left|{{\color{red}{v}}}\right| \right)}}{14} = \frac{u \operatorname{atan}{\left(u \right)}}{7} - \frac{\ln{\left(\left|{{\color{red}{\left(u^{2} + 1\right)}}}\right| \right)}}{14}$$

Ingat bahwa $$$u=7 t$$$:

$$- \frac{\ln{\left(1 + {\color{red}{u}}^{2} \right)}}{14} + \frac{{\color{red}{u}} \operatorname{atan}{\left({\color{red}{u}} \right)}}{7} = - \frac{\ln{\left(1 + {\color{red}{\left(7 t\right)}}^{2} \right)}}{14} + \frac{{\color{red}{\left(7 t\right)}} \operatorname{atan}{\left({\color{red}{\left(7 t\right)}} \right)}}{7}$$

Oleh karena itu,

$$\int{\operatorname{atan}{\left(7 t \right)} d t} = t \operatorname{atan}{\left(7 t \right)} - \frac{\ln{\left(49 t^{2} + 1 \right)}}{14}$$

Tambahkan konstanta integrasi:

$$\int{\operatorname{atan}{\left(7 t \right)} d t} = t \operatorname{atan}{\left(7 t \right)} - \frac{\ln{\left(49 t^{2} + 1 \right)}}{14}+C$$

$$$\int \operatorname{atan}{\left(7 t \right)}\, dt = \left(t \operatorname{atan}{\left(7 t \right)} - \frac{\ln\left(49 t^{2} + 1\right)}{14}\right) + C$$$A