Integral dari $$$\cot^{2}{\left(x + \frac{\pi}{4} \right)}$$$

Temukan $$$\int \cot^{2}{\left(x + \frac{\pi}{4} \right)}\, dx$$$.

Misalkan $$$u=x + \frac{\pi}{4}$$$.

Kemudian $$$du=\left(x + \frac{\pi}{4}\right)^{\prime }dx = 1 dx$$$ (langkah-langkah dapat dilihat di »), dan kita memperoleh $$$dx = du$$$.

Jadi,

$${\color{red}{\int{\cot^{2}{\left(x + \frac{\pi}{4} \right)} d x}}} = {\color{red}{\int{\cot^{2}{\left(u \right)} d u}}}$$

Misalkan $$$v=\cot{\left(u \right)}$$$.

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

Integralnya menjadi

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

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

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

Tulis ulang dan pisahkan pecahannya:

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

Integralkan suku demi suku:

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

Terapkan aturan konstanta $$$\int c\, dv = c v$$$ dengan $$$c=1$$$:

$$\int{\frac{1}{v^{2} + 1} d v} - {\color{red}{\int{1 d v}}} = \int{\frac{1}{v^{2} + 1} d v} - {\color{red}{v}}$$

Integral dari $$$\frac{1}{v^{2} + 1}$$$ adalah $$$\int{\frac{1}{v^{2} + 1} d v} = \operatorname{atan}{\left(v \right)}$$$:

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

Ingat bahwa $$$v=\cot{\left(u \right)}$$$:

$$\operatorname{atan}{\left({\color{red}{v}} \right)} - {\color{red}{v}} = \operatorname{atan}{\left({\color{red}{\cot{\left(u \right)}}} \right)} - {\color{red}{\cot{\left(u \right)}}}$$

Ingat bahwa $$$u=x + \frac{\pi}{4}$$$:

$$- \cot{\left({\color{red}{u}} \right)} + \operatorname{atan}{\left(\cot{\left({\color{red}{u}} \right)} \right)} = - \cot{\left({\color{red}{\left(x + \frac{\pi}{4}\right)}} \right)} + \operatorname{atan}{\left(\cot{\left({\color{red}{\left(x + \frac{\pi}{4}\right)}} \right)} \right)}$$

Oleh karena itu,

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

Tambahkan konstanta integrasi:

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

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