$$$\left(2 - 3 \sin^{2}{\left(x \right)}\right) \sin{\left(x \right)}$$$の積分

$$$\int \left(2 - 3 \sin^{2}{\left(x \right)}\right) \sin{\left(x \right)}\, dx$$$ を求めよ。

Expand the expression:

$${\color{red}{\int{\left(2 - 3 \sin^{2}{\left(x \right)}\right) \sin{\left(x \right)} d x}}} = {\color{red}{\int{\left(- 3 \sin^{3}{\left(x \right)} + 2 \sin{\left(x \right)}\right)d x}}}$$

項別に積分せよ:

$${\color{red}{\int{\left(- 3 \sin^{3}{\left(x \right)} + 2 \sin{\left(x \right)}\right)d x}}} = {\color{red}{\left(\int{2 \sin{\left(x \right)} d x} - \int{3 \sin^{3}{\left(x \right)} d x}\right)}}$$

定数倍の法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ を、$$$c=3$$$ と $$$f{\left(x \right)} = \sin^{3}{\left(x \right)}$$$ に対して適用する:

$$\int{2 \sin{\left(x \right)} d x} - {\color{red}{\int{3 \sin^{3}{\left(x \right)} d x}}} = \int{2 \sin{\left(x \right)} d x} - {\color{red}{\left(3 \int{\sin^{3}{\left(x \right)} d x}\right)}}$$

正弦を1つ取り出し、残りは余弦で表し、$$$\alpha=x$$$ に対する公式 $$$\sin^2\left(\alpha \right)=-\cos^2\left(\alpha \right)+1$$$ を用いよ。:

$$\int{2 \sin{\left(x \right)} d x} - 3 {\color{red}{\int{\sin^{3}{\left(x \right)} d x}}} = \int{2 \sin{\left(x \right)} d x} - 3 {\color{red}{\int{\left(1 - \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)} d x}}}$$

$$$u=\cos{\left(x \right)}$$$ とする。

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

積分は次のようになります

$$\int{2 \sin{\left(x \right)} d x} - 3 {\color{red}{\int{\left(1 - \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)} d x}}} = \int{2 \sin{\left(x \right)} d x} - 3 {\color{red}{\int{\left(u^{2} - 1\right)d u}}}$$

定数倍の法則 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ を、$$$c=-1$$$ と $$$f{\left(u \right)} = 1 - u^{2}$$$ に対して適用する:

$$\int{2 \sin{\left(x \right)} d x} - 3 {\color{red}{\int{\left(u^{2} - 1\right)d u}}} = \int{2 \sin{\left(x \right)} d x} - 3 {\color{red}{\left(- \int{\left(1 - u^{2}\right)d u}\right)}}$$

項別に積分せよ:

$$\int{2 \sin{\left(x \right)} d x} + 3 {\color{red}{\int{\left(1 - u^{2}\right)d u}}} = \int{2 \sin{\left(x \right)} d x} + 3 {\color{red}{\left(\int{1 d u} - \int{u^{2} d u}\right)}}$$

$$$c=1$$$ に対して定数則 $$$\int c\, du = c u$$$ を適用する:

$$\int{2 \sin{\left(x \right)} d x} - 3 \int{u^{2} d u} + 3 {\color{red}{\int{1 d u}}} = \int{2 \sin{\left(x \right)} d x} - 3 \int{u^{2} d u} + 3 {\color{red}{u}}$$

$$$n=2$$$ を用いて、べき乗の法則 $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ を適用します:

$$3 u + \int{2 \sin{\left(x \right)} d x} - 3 {\color{red}{\int{u^{2} d u}}}=3 u + \int{2 \sin{\left(x \right)} d x} - 3 {\color{red}{\frac{u^{1 + 2}}{1 + 2}}}=3 u + \int{2 \sin{\left(x \right)} d x} - 3 {\color{red}{\left(\frac{u^{3}}{3}\right)}}$$

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

$$\int{2 \sin{\left(x \right)} d x} + 3 {\color{red}{u}} - {\color{red}{u}}^{3} = \int{2 \sin{\left(x \right)} d x} + 3 {\color{red}{\cos{\left(x \right)}}} - {\color{red}{\cos{\left(x \right)}}}^{3}$$

定数倍の法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ を、$$$c=2$$$ と $$$f{\left(x \right)} = \sin{\left(x \right)}$$$ に対して適用する:

$$- \cos^{3}{\left(x \right)} + 3 \cos{\left(x \right)} + {\color{red}{\int{2 \sin{\left(x \right)} d x}}} = - \cos^{3}{\left(x \right)} + 3 \cos{\left(x \right)} + {\color{red}{\left(2 \int{\sin{\left(x \right)} d x}\right)}}$$

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

$$- \cos^{3}{\left(x \right)} + 3 \cos{\left(x \right)} + 2 {\color{red}{\int{\sin{\left(x \right)} d x}}} = - \cos^{3}{\left(x \right)} + 3 \cos{\left(x \right)} + 2 {\color{red}{\left(- \cos{\left(x \right)}\right)}}$$

したがって、

$$\int{\left(2 - 3 \sin^{2}{\left(x \right)}\right) \sin{\left(x \right)} d x} = - \cos^{3}{\left(x \right)} + \cos{\left(x \right)}$$

積分定数を加える:

$$\int{\left(2 - 3 \sin^{2}{\left(x \right)}\right) \sin{\left(x \right)} d x} = - \cos^{3}{\left(x \right)} + \cos{\left(x \right)}+C$$

$$$\int \left(2 - 3 \sin^{2}{\left(x \right)}\right) \sin{\left(x \right)}\, dx = \left(- \cos^{3}{\left(x \right)} + \cos{\left(x \right)}\right) + C$$$A