$$$\csc^{3}{\left(x \right)}$$$の積分

$$$\int \csc^{3}{\left(x \right)}\, dx$$$ を求めよ。

積分 $$$\int{\csc^{3}{\left(x \right)} d x}$$$ には、部分積分法$$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$を用いてください。

$$$\operatorname{u}=\csc{\left(x \right)}$$$ と $$$\operatorname{dv}=\csc^{2}{\left(x \right)} dx$$$ とする。

したがって、$$$\operatorname{du}=\left(\csc{\left(x \right)}\right)^{\prime }dx=- \cot{\left(x \right)} \csc{\left(x \right)} dx$$$（手順は»を参照）および$$$\operatorname{v}=\int{\csc^{2}{\left(x \right)} d x}=- \cot{\left(x \right)}$$$（手順は»を参照）。

したがって、

$$\int{\csc^{3}{\left(x \right)} d x}=\csc{\left(x \right)} \cdot \left(- \cot{\left(x \right)}\right)-\int{\left(- \cot{\left(x \right)}\right) \cdot \left(- \cot{\left(x \right)} \csc{\left(x \right)}\right) d x}=- \cot{\left(x \right)} \csc{\left(x \right)} - \int{\cot^{2}{\left(x \right)} \csc{\left(x \right)} d x}$$

公式$$$\cot^{2}{\left(x \right)} = \csc^{2}{\left(x \right)} - 1$$$を適用します:

$$- \cot{\left(x \right)} \csc{\left(x \right)} - \int{\cot^{2}{\left(x \right)} \csc{\left(x \right)} d x}=- \cot{\left(x \right)} \csc{\left(x \right)} - \int{\left(\csc^{2}{\left(x \right)} - 1\right) \csc{\left(x \right)} d x}$$

展開せよ:

$$- \cot{\left(x \right)} \csc{\left(x \right)} - \int{\left(\csc^{2}{\left(x \right)} - 1\right) \csc{\left(x \right)} d x}=- \cot{\left(x \right)} \csc{\left(x \right)} - \int{\left(\csc^{3}{\left(x \right)} - \csc{\left(x \right)}\right)d x}$$

差の積分は積分の差に等しい:

$$- \cot{\left(x \right)} \csc{\left(x \right)} - \int{\left(\csc^{3}{\left(x \right)} - \csc{\left(x \right)}\right)d x}=- \cot{\left(x \right)} \csc{\left(x \right)} + \int{\csc{\left(x \right)} d x} - \int{\csc^{3}{\left(x \right)} d x}$$

したがって、積分に関する次の簡単な線形方程式が得られます。

$${\color{red}{\int{\csc^{3}{\left(x \right)} d x}}}=- \cot{\left(x \right)} \csc{\left(x \right)} + \int{\csc{\left(x \right)} d x} - {\color{red}{\int{\csc^{3}{\left(x \right)} d x}}}$$

これを解くと、次のようになる。

$$\int{\csc^{3}{\left(x \right)} d x}=- \frac{\cot{\left(x \right)} \csc{\left(x \right)}}{2} + \frac{\int{\csc{\left(x \right)} d x}}{2}$$

余割関数を$$$\csc\left(x\right)=\frac{1}{\sin\left(x\right)}$$$として書き換えなさい:

$$- \frac{\cot{\left(x \right)} \csc{\left(x \right)}}{2} + \frac{{\color{red}{\int{\csc{\left(x \right)} d x}}}}{2} = - \frac{\cot{\left(x \right)} \csc{\left(x \right)}}{2} + \frac{{\color{red}{\int{\frac{1}{\sin{\left(x \right)}} d x}}}}{2}$$

二倍角の公式を用いて正弦を書き換える $$$\sin\left(x\right)=2\sin\left(\frac{x}{2}\right)\cos\left(\frac{x}{2}\right)$$$:

$$- \frac{\cot{\left(x \right)} \csc{\left(x \right)}}{2} + \frac{{\color{red}{\int{\frac{1}{\sin{\left(x \right)}} d x}}}}{2} = - \frac{\cot{\left(x \right)} \csc{\left(x \right)}}{2} + \frac{{\color{red}{\int{\frac{1}{2 \sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)}} d x}}}}{2}$$

分子と分母に$$$\sec^2\left(\frac{x}{2} \right)$$$を掛ける:

$$- \frac{\cot{\left(x \right)} \csc{\left(x \right)}}{2} + \frac{{\color{red}{\int{\frac{1}{2 \sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)}} d x}}}}{2} = - \frac{\cot{\left(x \right)} \csc{\left(x \right)}}{2} + \frac{{\color{red}{\int{\frac{\sec^{2}{\left(\frac{x}{2} \right)}}{2 \tan{\left(\frac{x}{2} \right)}} d x}}}}{2}$$

$$$u=\tan{\left(\frac{x}{2} \right)}$$$ とする。

すると $$$du=\left(\tan{\left(\frac{x}{2} \right)}\right)^{\prime }dx = \frac{\sec^{2}{\left(\frac{x}{2} \right)}}{2} dx$$$（手順は»で確認できます）、$$$\sec^{2}{\left(\frac{x}{2} \right)} dx = 2 du$$$ となります。

したがって、

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

$$$\frac{1}{u}$$$ の不定積分は $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$ です:

$$- \frac{\cot{\left(x \right)} \csc{\left(x \right)}}{2} + \frac{{\color{red}{\int{\frac{1}{u} d u}}}}{2} = - \frac{\cot{\left(x \right)} \csc{\left(x \right)}}{2} + \frac{{\color{red}{\ln{\left(\left|{u}\right| \right)}}}}{2}$$

次のことを思い出してください $$$u=\tan{\left(\frac{x}{2} \right)}$$$:

$$\frac{\ln{\left(\left|{{\color{red}{u}}}\right| \right)}}{2} - \frac{\cot{\left(x \right)} \csc{\left(x \right)}}{2} = \frac{\ln{\left(\left|{{\color{red}{\tan{\left(\frac{x}{2} \right)}}}}\right| \right)}}{2} - \frac{\cot{\left(x \right)} \csc{\left(x \right)}}{2}$$

したがって、

$$\int{\csc^{3}{\left(x \right)} d x} = \frac{\ln{\left(\left|{\tan{\left(\frac{x}{2} \right)}}\right| \right)}}{2} - \frac{\cot{\left(x \right)} \csc{\left(x \right)}}{2}$$

積分定数を加える:

$$\int{\csc^{3}{\left(x \right)} d x} = \frac{\ln{\left(\left|{\tan{\left(\frac{x}{2} \right)}}\right| \right)}}{2} - \frac{\cot{\left(x \right)} \csc{\left(x \right)}}{2}+C$$

$$$\int \csc^{3}{\left(x \right)}\, dx = \left(\frac{\ln\left(\left|{\tan{\left(\frac{x}{2} \right)}}\right|\right)}{2} - \frac{\cot{\left(x \right)} \csc{\left(x \right)}}{2}\right) + C$$$A