$$$5 \sin^{2}{\left(7 x \right)} \cos^{3}{\left(7 x \right)}$$$の積分

$$$\int 5 \sin^{2}{\left(7 x \right)} \cos^{3}{\left(7 x \right)}\, dx$$$ を求めよ。

冪低減公式 $$$\cos^{3}{\left(\alpha \right)} = \frac{3 \cos{\left(\alpha \right)}}{4} + \frac{\cos{\left(3 \alpha \right)}}{4}$$$ を $$$\alpha=7 x$$$ に適用する:

$${\color{red}{\int{5 \sin^{2}{\left(7 x \right)} \cos^{3}{\left(7 x \right)} d x}}} = {\color{red}{\int{\frac{5 \left(3 \cos{\left(7 x \right)} + \cos{\left(21 x \right)}\right) \sin^{2}{\left(7 x \right)}}{4} d x}}}$$

冪低減公式 $$$\sin^{2}{\left(\alpha \right)} = \frac{1}{2} - \frac{\cos{\left(2 \alpha \right)}}{2}$$$ を $$$\alpha=7 x$$$ に適用する:

$${\color{red}{\int{\frac{5 \left(3 \cos{\left(7 x \right)} + \cos{\left(21 x \right)}\right) \sin^{2}{\left(7 x \right)}}{4} d x}}} = {\color{red}{\int{\frac{5 \left(1 - \cos{\left(14 x \right)}\right) \left(3 \cos{\left(7 x \right)} + \cos{\left(21 x \right)}\right)}{8} d x}}}$$

定数倍の法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ を、$$$c=\frac{1}{8}$$$ と $$$f{\left(x \right)} = 5 \left(1 - \cos{\left(14 x \right)}\right) \left(3 \cos{\left(7 x \right)} + \cos{\left(21 x \right)}\right)$$$ に対して適用する:

$${\color{red}{\int{\frac{5 \left(1 - \cos{\left(14 x \right)}\right) \left(3 \cos{\left(7 x \right)} + \cos{\left(21 x \right)}\right)}{8} d x}}} = {\color{red}{\left(\frac{\int{5 \left(1 - \cos{\left(14 x \right)}\right) \left(3 \cos{\left(7 x \right)} + \cos{\left(21 x \right)}\right) d x}}{8}\right)}}$$

Expand the expression:

$$\frac{{\color{red}{\int{5 \left(1 - \cos{\left(14 x \right)}\right) \left(3 \cos{\left(7 x \right)} + \cos{\left(21 x \right)}\right) d x}}}}{8} = \frac{{\color{red}{\int{\left(- 15 \cos{\left(7 x \right)} \cos{\left(14 x \right)} + 15 \cos{\left(7 x \right)} - 5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} + 5 \cos{\left(21 x \right)}\right)d x}}}}{8}$$

項別に積分せよ:

$$\frac{{\color{red}{\int{\left(- 15 \cos{\left(7 x \right)} \cos{\left(14 x \right)} + 15 \cos{\left(7 x \right)} - 5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} + 5 \cos{\left(21 x \right)}\right)d x}}}}{8} = \frac{{\color{red}{\left(- \int{15 \cos{\left(7 x \right)} \cos{\left(14 x \right)} d x} - \int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x} + \int{15 \cos{\left(7 x \right)} d x} + \int{5 \cos{\left(21 x \right)} d x}\right)}}}{8}$$

定数倍の法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ を、$$$c=5$$$ と $$$f{\left(x \right)} = \cos{\left(21 x \right)}$$$ に対して適用する:

$$- \frac{\int{15 \cos{\left(7 x \right)} \cos{\left(14 x \right)} d x}}{8} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} + \frac{\int{15 \cos{\left(7 x \right)} d x}}{8} + \frac{{\color{red}{\int{5 \cos{\left(21 x \right)} d x}}}}{8} = - \frac{\int{15 \cos{\left(7 x \right)} \cos{\left(14 x \right)} d x}}{8} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} + \frac{\int{15 \cos{\left(7 x \right)} d x}}{8} + \frac{{\color{red}{\left(5 \int{\cos{\left(21 x \right)} d x}\right)}}}{8}$$

$$$u=21 x$$$ とする。

すると $$$du=\left(21 x\right)^{\prime }dx = 21 dx$$$（手順は»で確認できます）、$$$dx = \frac{du}{21}$$$ となります。

したがって、

$$- \frac{\int{15 \cos{\left(7 x \right)} \cos{\left(14 x \right)} d x}}{8} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} + \frac{\int{15 \cos{\left(7 x \right)} d x}}{8} + \frac{5 {\color{red}{\int{\cos{\left(21 x \right)} d x}}}}{8} = - \frac{\int{15 \cos{\left(7 x \right)} \cos{\left(14 x \right)} d x}}{8} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} + \frac{\int{15 \cos{\left(7 x \right)} d x}}{8} + \frac{5 {\color{red}{\int{\frac{\cos{\left(u \right)}}{21} d u}}}}{8}$$

定数倍の法則 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ を、$$$c=\frac{1}{21}$$$ と $$$f{\left(u \right)} = \cos{\left(u \right)}$$$ に対して適用する:

$$- \frac{\int{15 \cos{\left(7 x \right)} \cos{\left(14 x \right)} d x}}{8} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} + \frac{\int{15 \cos{\left(7 x \right)} d x}}{8} + \frac{5 {\color{red}{\int{\frac{\cos{\left(u \right)}}{21} d u}}}}{8} = - \frac{\int{15 \cos{\left(7 x \right)} \cos{\left(14 x \right)} d x}}{8} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} + \frac{\int{15 \cos{\left(7 x \right)} d x}}{8} + \frac{5 {\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{21}\right)}}}{8}$$

余弦の積分は$$$\int{\cos{\left(u \right)} d u} = \sin{\left(u \right)}$$$:

$$- \frac{\int{15 \cos{\left(7 x \right)} \cos{\left(14 x \right)} d x}}{8} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} + \frac{\int{15 \cos{\left(7 x \right)} d x}}{8} + \frac{5 {\color{red}{\int{\cos{\left(u \right)} d u}}}}{168} = - \frac{\int{15 \cos{\left(7 x \right)} \cos{\left(14 x \right)} d x}}{8} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} + \frac{\int{15 \cos{\left(7 x \right)} d x}}{8} + \frac{5 {\color{red}{\sin{\left(u \right)}}}}{168}$$

次のことを思い出してください $$$u=21 x$$$:

$$- \frac{\int{15 \cos{\left(7 x \right)} \cos{\left(14 x \right)} d x}}{8} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} + \frac{\int{15 \cos{\left(7 x \right)} d x}}{8} + \frac{5 \sin{\left({\color{red}{u}} \right)}}{168} = - \frac{\int{15 \cos{\left(7 x \right)} \cos{\left(14 x \right)} d x}}{8} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} + \frac{\int{15 \cos{\left(7 x \right)} d x}}{8} + \frac{5 \sin{\left({\color{red}{\left(21 x\right)}} \right)}}{168}$$

定数倍の法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ を、$$$c=15$$$ と $$$f{\left(x \right)} = \cos{\left(7 x \right)}$$$ に対して適用する:

$$\frac{5 \sin{\left(21 x \right)}}{168} - \frac{\int{15 \cos{\left(7 x \right)} \cos{\left(14 x \right)} d x}}{8} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} + \frac{{\color{red}{\int{15 \cos{\left(7 x \right)} d x}}}}{8} = \frac{5 \sin{\left(21 x \right)}}{168} - \frac{\int{15 \cos{\left(7 x \right)} \cos{\left(14 x \right)} d x}}{8} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} + \frac{{\color{red}{\left(15 \int{\cos{\left(7 x \right)} d x}\right)}}}{8}$$

$$$u=7 x$$$ とする。

すると $$$du=\left(7 x\right)^{\prime }dx = 7 dx$$$（手順は»で確認できます）、$$$dx = \frac{du}{7}$$$ となります。

この積分は次のように書き換えられる

$$\frac{5 \sin{\left(21 x \right)}}{168} - \frac{\int{15 \cos{\left(7 x \right)} \cos{\left(14 x \right)} d x}}{8} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} + \frac{15 {\color{red}{\int{\cos{\left(7 x \right)} d x}}}}{8} = \frac{5 \sin{\left(21 x \right)}}{168} - \frac{\int{15 \cos{\left(7 x \right)} \cos{\left(14 x \right)} d x}}{8} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} + \frac{15 {\color{red}{\int{\frac{\cos{\left(u \right)}}{7} d u}}}}{8}$$

定数倍の法則 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ を、$$$c=\frac{1}{7}$$$ と $$$f{\left(u \right)} = \cos{\left(u \right)}$$$ に対して適用する:

$$\frac{5 \sin{\left(21 x \right)}}{168} - \frac{\int{15 \cos{\left(7 x \right)} \cos{\left(14 x \right)} d x}}{8} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} + \frac{15 {\color{red}{\int{\frac{\cos{\left(u \right)}}{7} d u}}}}{8} = \frac{5 \sin{\left(21 x \right)}}{168} - \frac{\int{15 \cos{\left(7 x \right)} \cos{\left(14 x \right)} d x}}{8} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} + \frac{15 {\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{7}\right)}}}{8}$$

余弦の積分は$$$\int{\cos{\left(u \right)} d u} = \sin{\left(u \right)}$$$:

$$\frac{5 \sin{\left(21 x \right)}}{168} - \frac{\int{15 \cos{\left(7 x \right)} \cos{\left(14 x \right)} d x}}{8} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} + \frac{15 {\color{red}{\int{\cos{\left(u \right)} d u}}}}{56} = \frac{5 \sin{\left(21 x \right)}}{168} - \frac{\int{15 \cos{\left(7 x \right)} \cos{\left(14 x \right)} d x}}{8} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} + \frac{15 {\color{red}{\sin{\left(u \right)}}}}{56}$$

次のことを思い出してください $$$u=7 x$$$:

$$\frac{5 \sin{\left(21 x \right)}}{168} - \frac{\int{15 \cos{\left(7 x \right)} \cos{\left(14 x \right)} d x}}{8} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} + \frac{15 \sin{\left({\color{red}{u}} \right)}}{56} = \frac{5 \sin{\left(21 x \right)}}{168} - \frac{\int{15 \cos{\left(7 x \right)} \cos{\left(14 x \right)} d x}}{8} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} + \frac{15 \sin{\left({\color{red}{\left(7 x\right)}} \right)}}{56}$$

$$$\cos\left(\alpha \right)\cos\left(\beta \right)=\frac{1}{2} \cos\left(\alpha-\beta \right)+\frac{1}{2} \cos\left(\alpha+\beta \right)$$$ の公式を用い、$$$\alpha=7 x$$$ および $$$\beta=14 x$$$ を用いて $$$\cos\left(7 x \right)\cos\left(14 x \right)$$$ を変形せよ:

$$\frac{15 \sin{\left(7 x \right)}}{56} + \frac{5 \sin{\left(21 x \right)}}{168} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} - \frac{{\color{red}{\int{15 \cos{\left(7 x \right)} \cos{\left(14 x \right)} d x}}}}{8} = \frac{15 \sin{\left(7 x \right)}}{56} + \frac{5 \sin{\left(21 x \right)}}{168} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} - \frac{{\color{red}{\int{\left(\frac{15 \cos{\left(7 x \right)}}{2} + \frac{15 \cos{\left(21 x \right)}}{2}\right)d x}}}}{8}$$

定数倍の法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ を、$$$c=\frac{1}{2}$$$ と $$$f{\left(x \right)} = 15 \cos{\left(7 x \right)} + 15 \cos{\left(21 x \right)}$$$ に対して適用する:

$$\frac{15 \sin{\left(7 x \right)}}{56} + \frac{5 \sin{\left(21 x \right)}}{168} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} - \frac{{\color{red}{\int{\left(\frac{15 \cos{\left(7 x \right)}}{2} + \frac{15 \cos{\left(21 x \right)}}{2}\right)d x}}}}{8} = \frac{15 \sin{\left(7 x \right)}}{56} + \frac{5 \sin{\left(21 x \right)}}{168} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} - \frac{{\color{red}{\left(\frac{\int{\left(15 \cos{\left(7 x \right)} + 15 \cos{\left(21 x \right)}\right)d x}}{2}\right)}}}{8}$$

項別に積分せよ:

$$\frac{15 \sin{\left(7 x \right)}}{56} + \frac{5 \sin{\left(21 x \right)}}{168} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} - \frac{{\color{red}{\int{\left(15 \cos{\left(7 x \right)} + 15 \cos{\left(21 x \right)}\right)d x}}}}{16} = \frac{15 \sin{\left(7 x \right)}}{56} + \frac{5 \sin{\left(21 x \right)}}{168} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} - \frac{{\color{red}{\left(\int{15 \cos{\left(7 x \right)} d x} + \int{15 \cos{\left(21 x \right)} d x}\right)}}}{16}$$

定数倍の法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ を、$$$c=15$$$ と $$$f{\left(x \right)} = \cos{\left(7 x \right)}$$$ に対して適用する:

$$\frac{15 \sin{\left(7 x \right)}}{56} + \frac{5 \sin{\left(21 x \right)}}{168} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} - \frac{\int{15 \cos{\left(21 x \right)} d x}}{16} - \frac{{\color{red}{\int{15 \cos{\left(7 x \right)} d x}}}}{16} = \frac{15 \sin{\left(7 x \right)}}{56} + \frac{5 \sin{\left(21 x \right)}}{168} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} - \frac{\int{15 \cos{\left(21 x \right)} d x}}{16} - \frac{{\color{red}{\left(15 \int{\cos{\left(7 x \right)} d x}\right)}}}{16}$$

積分 $$$\int{\cos{\left(7 x \right)} d x}$$$ はすでに計算されています:

$$\int{\cos{\left(7 x \right)} d x} = \frac{\sin{\left(7 x \right)}}{7}$$

したがって、

$$\frac{15 \sin{\left(7 x \right)}}{56} + \frac{5 \sin{\left(21 x \right)}}{168} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} - \frac{\int{15 \cos{\left(21 x \right)} d x}}{16} - \frac{15 {\color{red}{\int{\cos{\left(7 x \right)} d x}}}}{16} = \frac{15 \sin{\left(7 x \right)}}{56} + \frac{5 \sin{\left(21 x \right)}}{168} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} - \frac{\int{15 \cos{\left(21 x \right)} d x}}{16} - \frac{15 {\color{red}{\left(\frac{\sin{\left(7 x \right)}}{7}\right)}}}{16}$$

定数倍の法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ を、$$$c=15$$$ と $$$f{\left(x \right)} = \cos{\left(21 x \right)}$$$ に対して適用する:

$$\frac{15 \sin{\left(7 x \right)}}{112} + \frac{5 \sin{\left(21 x \right)}}{168} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} - \frac{{\color{red}{\int{15 \cos{\left(21 x \right)} d x}}}}{16} = \frac{15 \sin{\left(7 x \right)}}{112} + \frac{5 \sin{\left(21 x \right)}}{168} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} - \frac{{\color{red}{\left(15 \int{\cos{\left(21 x \right)} d x}\right)}}}{16}$$

積分 $$$\int{\cos{\left(21 x \right)} d x}$$$ はすでに計算されています:

$$\int{\cos{\left(21 x \right)} d x} = \frac{\sin{\left(21 x \right)}}{21}$$

したがって、

$$\frac{15 \sin{\left(7 x \right)}}{112} + \frac{5 \sin{\left(21 x \right)}}{168} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} - \frac{15 {\color{red}{\int{\cos{\left(21 x \right)} d x}}}}{16} = \frac{15 \sin{\left(7 x \right)}}{112} + \frac{5 \sin{\left(21 x \right)}}{168} - \frac{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}{8} - \frac{15 {\color{red}{\left(\frac{\sin{\left(21 x \right)}}{21}\right)}}}{16}$$

$$$\cos\left(\alpha \right)\cos\left(\beta \right)=\frac{1}{2} \cos\left(\alpha-\beta \right)+\frac{1}{2} \cos\left(\alpha+\beta \right)$$$ の公式を用い、$$$\alpha=14 x$$$ および $$$\beta=21 x$$$ を用いて $$$\cos\left(14 x \right)\cos\left(21 x \right)$$$ を変形せよ:

$$\frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{{\color{red}{\int{5 \cos{\left(14 x \right)} \cos{\left(21 x \right)} d x}}}}{8} = \frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{{\color{red}{\int{\left(\frac{5 \cos{\left(7 x \right)}}{2} + \frac{5 \cos{\left(35 x \right)}}{2}\right)d x}}}}{8}$$

定数倍の法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ を、$$$c=\frac{1}{2}$$$ と $$$f{\left(x \right)} = 5 \cos{\left(7 x \right)} + 5 \cos{\left(35 x \right)}$$$ に対して適用する:

$$\frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{{\color{red}{\int{\left(\frac{5 \cos{\left(7 x \right)}}{2} + \frac{5 \cos{\left(35 x \right)}}{2}\right)d x}}}}{8} = \frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{{\color{red}{\left(\frac{\int{\left(5 \cos{\left(7 x \right)} + 5 \cos{\left(35 x \right)}\right)d x}}{2}\right)}}}{8}$$

被積分関数を書き換える:

$$\frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{{\color{red}{\int{\left(5 \cos{\left(7 x \right)} + 5 \cos{\left(35 x \right)}\right)d x}}}}{16} = \frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{{\color{red}{\int{5 \left(\cos{\left(7 x \right)} + \cos{\left(35 x \right)}\right) d x}}}}{16}$$

被積分関数を簡単化する:

$$\frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{{\color{red}{\int{5 \left(\cos{\left(7 x \right)} + \cos{\left(35 x \right)}\right) d x}}}}{16} = \frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{{\color{red}{\int{\left(5 \cos{\left(7 x \right)} + 5 \cos{\left(35 x \right)}\right)d x}}}}{16}$$

定数倍の法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ を、$$$c=5$$$ と $$$f{\left(x \right)} = \cos{\left(7 x \right)} + \cos{\left(35 x \right)}$$$ に対して適用する:

$$\frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{{\color{red}{\int{\left(5 \cos{\left(7 x \right)} + 5 \cos{\left(35 x \right)}\right)d x}}}}{16} = \frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{{\color{red}{\left(5 \int{\left(\cos{\left(7 x \right)} + \cos{\left(35 x \right)}\right)d x}\right)}}}{16}$$

$$$w=7 x$$$ とする。

すると $$$dw=\left(7 x\right)^{\prime }dx = 7 dx$$$（手順は»で確認できます）、$$$dx = \frac{dw}{7}$$$ となります。

したがって、

$$\frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{5 {\color{red}{\int{\left(\cos{\left(7 x \right)} + \cos{\left(35 x \right)}\right)d x}}}}{16} = \frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{5 {\color{red}{\int{\left(\frac{\cos{\left(w \right)}}{7} + \frac{\cos{\left(5 w \right)}}{7}\right)d w}}}}{16}$$

定数倍の法則 $$$\int c f{\left(w \right)}\, dw = c \int f{\left(w \right)}\, dw$$$ を、$$$c=\frac{1}{7}$$$ と $$$f{\left(w \right)} = \cos{\left(w \right)} + \cos{\left(5 w \right)}$$$ に対して適用する:

$$\frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{5 {\color{red}{\int{\left(\frac{\cos{\left(w \right)}}{7} + \frac{\cos{\left(5 w \right)}}{7}\right)d w}}}}{16} = \frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{5 {\color{red}{\left(\frac{\int{\left(\cos{\left(w \right)} + \cos{\left(5 w \right)}\right)d w}}{7}\right)}}}{16}$$

項別に積分せよ:

$$\frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{5 {\color{red}{\int{\left(\cos{\left(w \right)} + \cos{\left(5 w \right)}\right)d w}}}}{112} = \frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{5 {\color{red}{\left(\int{\cos{\left(w \right)} d w} + \int{\cos{\left(5 w \right)} d w}\right)}}}{112}$$

余弦の積分は$$$\int{\cos{\left(w \right)} d w} = \sin{\left(w \right)}$$$:

$$\frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{5 \int{\cos{\left(5 w \right)} d w}}{112} - \frac{5 {\color{red}{\int{\cos{\left(w \right)} d w}}}}{112} = \frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{5 \int{\cos{\left(5 w \right)} d w}}{112} - \frac{5 {\color{red}{\sin{\left(w \right)}}}}{112}$$

$$$\theta=5 w$$$ とする。

すると $$$d\theta=\left(5 w\right)^{\prime }dw = 5 dw$$$（手順は»で確認できます）、$$$dw = \frac{d\theta}{5}$$$ となります。

したがって、

$$- \frac{5 \sin{\left(w \right)}}{112} + \frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{5 {\color{red}{\int{\cos{\left(5 w \right)} d w}}}}{112} = - \frac{5 \sin{\left(w \right)}}{112} + \frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{5 {\color{red}{\int{\frac{\cos{\left(\theta \right)}}{5} d \theta}}}}{112}$$

定数倍の法則 $$$\int c f{\left(\theta \right)}\, d\theta = c \int f{\left(\theta \right)}\, d\theta$$$ を、$$$c=\frac{1}{5}$$$ と $$$f{\left(\theta \right)} = \cos{\left(\theta \right)}$$$ に対して適用する:

$$- \frac{5 \sin{\left(w \right)}}{112} + \frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{5 {\color{red}{\int{\frac{\cos{\left(\theta \right)}}{5} d \theta}}}}{112} = - \frac{5 \sin{\left(w \right)}}{112} + \frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{5 {\color{red}{\left(\frac{\int{\cos{\left(\theta \right)} d \theta}}{5}\right)}}}{112}$$

余弦の積分は$$$\int{\cos{\left(\theta \right)} d \theta} = \sin{\left(\theta \right)}$$$:

$$- \frac{5 \sin{\left(w \right)}}{112} + \frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{{\color{red}{\int{\cos{\left(\theta \right)} d \theta}}}}{112} = - \frac{5 \sin{\left(w \right)}}{112} + \frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{{\color{red}{\sin{\left(\theta \right)}}}}{112}$$

次のことを思い出してください $$$\theta=5 w$$$:

$$- \frac{5 \sin{\left(w \right)}}{112} + \frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{\sin{\left({\color{red}{\theta}} \right)}}{112} = - \frac{5 \sin{\left(w \right)}}{112} + \frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{\sin{\left({\color{red}{\left(5 w\right)}} \right)}}{112}$$

次のことを思い出してください $$$w=7 x$$$:

$$\frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{5 \sin{\left({\color{red}{w}} \right)}}{112} - \frac{\sin{\left(5 {\color{red}{w}} \right)}}{112} = \frac{15 \sin{\left(7 x \right)}}{112} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{5 \sin{\left({\color{red}{\left(7 x\right)}} \right)}}{112} - \frac{\sin{\left(5 {\color{red}{\left(7 x\right)}} \right)}}{112}$$

したがって、

$$\int{5 \sin^{2}{\left(7 x \right)} \cos^{3}{\left(7 x \right)} d x} = \frac{5 \sin{\left(7 x \right)}}{56} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{\sin{\left(35 x \right)}}{112}$$

積分定数を加える:

$$\int{5 \sin^{2}{\left(7 x \right)} \cos^{3}{\left(7 x \right)} d x} = \frac{5 \sin{\left(7 x \right)}}{56} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{\sin{\left(35 x \right)}}{112}+C$$

$$$\int 5 \sin^{2}{\left(7 x \right)} \cos^{3}{\left(7 x \right)}\, dx = \left(\frac{5 \sin{\left(7 x \right)}}{56} - \frac{5 \sin{\left(21 x \right)}}{336} - \frac{\sin{\left(35 x \right)}}{112}\right) + C$$$A