$$$3 \sin^{2}{\left(t \right)} + 18 \sin{\left(t \right)} + \frac{86 \cos{\left(t \right)}}{21} + 1$$$の積分

$$$\int \left(3 \sin^{2}{\left(t \right)} + 18 \sin{\left(t \right)} + \frac{86 \cos{\left(t \right)}}{21} + 1\right)\, dt$$$ を求めよ。

項別に積分せよ:

$${\color{red}{\int{\left(3 \sin^{2}{\left(t \right)} + 18 \sin{\left(t \right)} + \frac{86 \cos{\left(t \right)}}{21} + 1\right)d t}}} = {\color{red}{\left(\int{1 d t} + \int{18 \sin{\left(t \right)} d t} + \int{3 \sin^{2}{\left(t \right)} d t} + \int{\frac{86 \cos{\left(t \right)}}{21} d t}\right)}}$$

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

$$\int{18 \sin{\left(t \right)} d t} + \int{3 \sin^{2}{\left(t \right)} d t} + \int{\frac{86 \cos{\left(t \right)}}{21} d t} + {\color{red}{\int{1 d t}}} = \int{18 \sin{\left(t \right)} d t} + \int{3 \sin^{2}{\left(t \right)} d t} + \int{\frac{86 \cos{\left(t \right)}}{21} d t} + {\color{red}{t}}$$

定数倍の法則 $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$ を、$$$c=3$$$ と $$$f{\left(t \right)} = \sin^{2}{\left(t \right)}$$$ に対して適用する:

$$t + \int{18 \sin{\left(t \right)} d t} + \int{\frac{86 \cos{\left(t \right)}}{21} d t} + {\color{red}{\int{3 \sin^{2}{\left(t \right)} d t}}} = t + \int{18 \sin{\left(t \right)} d t} + \int{\frac{86 \cos{\left(t \right)}}{21} d t} + {\color{red}{\left(3 \int{\sin^{2}{\left(t \right)} d t}\right)}}$$

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

$$t + \int{18 \sin{\left(t \right)} d t} + \int{\frac{86 \cos{\left(t \right)}}{21} d t} + 3 {\color{red}{\int{\sin^{2}{\left(t \right)} d t}}} = t + \int{18 \sin{\left(t \right)} d t} + \int{\frac{86 \cos{\left(t \right)}}{21} d t} + 3 {\color{red}{\int{\left(\frac{1}{2} - \frac{\cos{\left(2 t \right)}}{2}\right)d t}}}$$

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

$$t + \int{18 \sin{\left(t \right)} d t} + \int{\frac{86 \cos{\left(t \right)}}{21} d t} + 3 {\color{red}{\int{\left(\frac{1}{2} - \frac{\cos{\left(2 t \right)}}{2}\right)d t}}} = t + \int{18 \sin{\left(t \right)} d t} + \int{\frac{86 \cos{\left(t \right)}}{21} d t} + 3 {\color{red}{\left(\frac{\int{\left(1 - \cos{\left(2 t \right)}\right)d t}}{2}\right)}}$$

項別に積分せよ:

$$t + \int{18 \sin{\left(t \right)} d t} + \int{\frac{86 \cos{\left(t \right)}}{21} d t} + \frac{3 {\color{red}{\int{\left(1 - \cos{\left(2 t \right)}\right)d t}}}}{2} = t + \int{18 \sin{\left(t \right)} d t} + \int{\frac{86 \cos{\left(t \right)}}{21} d t} + \frac{3 {\color{red}{\left(\int{1 d t} - \int{\cos{\left(2 t \right)} d t}\right)}}}{2}$$

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

$$t + \int{18 \sin{\left(t \right)} d t} + \int{\frac{86 \cos{\left(t \right)}}{21} d t} - \frac{3 \int{\cos{\left(2 t \right)} d t}}{2} + \frac{3 {\color{red}{\int{1 d t}}}}{2} = t + \int{18 \sin{\left(t \right)} d t} + \int{\frac{86 \cos{\left(t \right)}}{21} d t} - \frac{3 \int{\cos{\left(2 t \right)} d t}}{2} + \frac{3 {\color{red}{t}}}{2}$$

$$$u=2 t$$$ とする。

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

この積分は次のように書き換えられる

$$\frac{5 t}{2} + \int{18 \sin{\left(t \right)} d t} + \int{\frac{86 \cos{\left(t \right)}}{21} d t} - \frac{3 {\color{red}{\int{\cos{\left(2 t \right)} d t}}}}{2} = \frac{5 t}{2} + \int{18 \sin{\left(t \right)} d t} + \int{\frac{86 \cos{\left(t \right)}}{21} d t} - \frac{3 {\color{red}{\int{\frac{\cos{\left(u \right)}}{2} d u}}}}{2}$$

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

$$\frac{5 t}{2} + \int{18 \sin{\left(t \right)} d t} + \int{\frac{86 \cos{\left(t \right)}}{21} d t} - \frac{3 {\color{red}{\int{\frac{\cos{\left(u \right)}}{2} d u}}}}{2} = \frac{5 t}{2} + \int{18 \sin{\left(t \right)} d t} + \int{\frac{86 \cos{\left(t \right)}}{21} d t} - \frac{3 {\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{2}\right)}}}{2}$$

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

$$\frac{5 t}{2} + \int{18 \sin{\left(t \right)} d t} + \int{\frac{86 \cos{\left(t \right)}}{21} d t} - \frac{3 {\color{red}{\int{\cos{\left(u \right)} d u}}}}{4} = \frac{5 t}{2} + \int{18 \sin{\left(t \right)} d t} + \int{\frac{86 \cos{\left(t \right)}}{21} d t} - \frac{3 {\color{red}{\sin{\left(u \right)}}}}{4}$$

次のことを思い出してください $$$u=2 t$$$:

$$\frac{5 t}{2} + \int{18 \sin{\left(t \right)} d t} + \int{\frac{86 \cos{\left(t \right)}}{21} d t} - \frac{3 \sin{\left({\color{red}{u}} \right)}}{4} = \frac{5 t}{2} + \int{18 \sin{\left(t \right)} d t} + \int{\frac{86 \cos{\left(t \right)}}{21} d t} - \frac{3 \sin{\left({\color{red}{\left(2 t\right)}} \right)}}{4}$$

定数倍の法則 $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$ を、$$$c=18$$$ と $$$f{\left(t \right)} = \sin{\left(t \right)}$$$ に対して適用する:

$$\frac{5 t}{2} - \frac{3 \sin{\left(2 t \right)}}{4} + \int{\frac{86 \cos{\left(t \right)}}{21} d t} + {\color{red}{\int{18 \sin{\left(t \right)} d t}}} = \frac{5 t}{2} - \frac{3 \sin{\left(2 t \right)}}{4} + \int{\frac{86 \cos{\left(t \right)}}{21} d t} + {\color{red}{\left(18 \int{\sin{\left(t \right)} d t}\right)}}$$

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

$$\frac{5 t}{2} - \frac{3 \sin{\left(2 t \right)}}{4} + \int{\frac{86 \cos{\left(t \right)}}{21} d t} + 18 {\color{red}{\int{\sin{\left(t \right)} d t}}} = \frac{5 t}{2} - \frac{3 \sin{\left(2 t \right)}}{4} + \int{\frac{86 \cos{\left(t \right)}}{21} d t} + 18 {\color{red}{\left(- \cos{\left(t \right)}\right)}}$$

定数倍の法則 $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$ を、$$$c=\frac{86}{21}$$$ と $$$f{\left(t \right)} = \cos{\left(t \right)}$$$ に対して適用する:

$$\frac{5 t}{2} - \frac{3 \sin{\left(2 t \right)}}{4} - 18 \cos{\left(t \right)} + {\color{red}{\int{\frac{86 \cos{\left(t \right)}}{21} d t}}} = \frac{5 t}{2} - \frac{3 \sin{\left(2 t \right)}}{4} - 18 \cos{\left(t \right)} + {\color{red}{\left(\frac{86 \int{\cos{\left(t \right)} d t}}{21}\right)}}$$

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

$$\frac{5 t}{2} - \frac{3 \sin{\left(2 t \right)}}{4} - 18 \cos{\left(t \right)} + \frac{86 {\color{red}{\int{\cos{\left(t \right)} d t}}}}{21} = \frac{5 t}{2} - \frac{3 \sin{\left(2 t \right)}}{4} - 18 \cos{\left(t \right)} + \frac{86 {\color{red}{\sin{\left(t \right)}}}}{21}$$

したがって、

$$\int{\left(3 \sin^{2}{\left(t \right)} + 18 \sin{\left(t \right)} + \frac{86 \cos{\left(t \right)}}{21} + 1\right)d t} = \frac{5 t}{2} + \frac{86 \sin{\left(t \right)}}{21} - \frac{3 \sin{\left(2 t \right)}}{4} - 18 \cos{\left(t \right)}$$

簡単化せよ:

$$\int{\left(3 \sin^{2}{\left(t \right)} + 18 \sin{\left(t \right)} + \frac{86 \cos{\left(t \right)}}{21} + 1\right)d t} = \frac{210 t + 344 \sin{\left(t \right)} - 63 \sin{\left(2 t \right)} - 1512 \cos{\left(t \right)}}{84}$$

積分定数を加える:

$$\int{\left(3 \sin^{2}{\left(t \right)} + 18 \sin{\left(t \right)} + \frac{86 \cos{\left(t \right)}}{21} + 1\right)d t} = \frac{210 t + 344 \sin{\left(t \right)} - 63 \sin{\left(2 t \right)} - 1512 \cos{\left(t \right)}}{84}+C$$

$$$\int \left(3 \sin^{2}{\left(t \right)} + 18 \sin{\left(t \right)} + \frac{86 \cos{\left(t \right)}}{21} + 1\right)\, dt = \frac{210 t + 344 \sin{\left(t \right)} - 63 \sin{\left(2 t \right)} - 1512 \cos{\left(t \right)}}{84} + C$$$A