$$$\frac{- \sin^{2}{\left(x \right)} + \cos{\left(x \right)}}{\sin{\left(x \right)}}$$$の積分

$$$\int \frac{- \sin^{2}{\left(x \right)} + \cos{\left(x \right)}}{\sin{\left(x \right)}}\, dx$$$ を求めよ。

Expand the expression:

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

項別に積分せよ:

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

正弦関数の不定積分は$$$\int{\sin{\left(x \right)} d x} = - \cos{\left(x \right)}$$$です:

$$\int{\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} d x} - {\color{red}{\int{\sin{\left(x \right)} d x}}} = \int{\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} d x} - {\color{red}{\left(- \cos{\left(x \right)}\right)}}$$

$$$u=\sin{\left(x \right)}$$$ とする。

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

この積分は次のように書き換えられる

$$\cos{\left(x \right)} + {\color{red}{\int{\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} d x}}} = \cos{\left(x \right)} + {\color{red}{\int{\frac{1}{u} d u}}}$$

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

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

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

$$\ln{\left(\left|{{\color{red}{u}}}\right| \right)} + \cos{\left(x \right)} = \ln{\left(\left|{{\color{red}{\sin{\left(x \right)}}}}\right| \right)} + \cos{\left(x \right)}$$

したがって、

$$\int{\frac{- \sin^{2}{\left(x \right)} + \cos{\left(x \right)}}{\sin{\left(x \right)}} d x} = \ln{\left(\left|{\sin{\left(x \right)}}\right| \right)} + \cos{\left(x \right)}$$

積分定数を加える:

$$\int{\frac{- \sin^{2}{\left(x \right)} + \cos{\left(x \right)}}{\sin{\left(x \right)}} d x} = \ln{\left(\left|{\sin{\left(x \right)}}\right| \right)} + \cos{\left(x \right)}+C$$

$$$\int \frac{- \sin^{2}{\left(x \right)} + \cos{\left(x \right)}}{\sin{\left(x \right)}}\, dx = \left(\ln\left(\left|{\sin{\left(x \right)}}\right|\right) + \cos{\left(x \right)}\right) + C$$$A