$$$\frac{1003605945944011425233769242881280649744658171441 t}{1000000000000000000000000000000000000000000000000}$$$の積分
関連する計算機: 定積分・広義積分計算機
入力内容
$$$\int \frac{1003605945944011425233769242881280649744658171441 t}{1000000000000000000000000000000000000000000000000}\, dt$$$ を求めよ。
解答
定数倍の法則 $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$ を、$$$c=\frac{1003605945944011425233769242881280649744658171441}{1000000000000000000000000000000000000000000000000}$$$ と $$$f{\left(t \right)} = t$$$ に対して適用する:
$${\color{red}{\int{\frac{1003605945944011425233769242881280649744658171441 t}{1000000000000000000000000000000000000000000000000} d t}}} = {\color{red}{\left(\frac{1003605945944011425233769242881280649744658171441 \int{t d t}}{1000000000000000000000000000000000000000000000000}\right)}}$$
$$$n=1$$$ を用いて、べき乗の法則 $$$\int t^{n}\, dt = \frac{t^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ を適用します:
$$\frac{1003605945944011425233769242881280649744658171441 {\color{red}{\int{t d t}}}}{1000000000000000000000000000000000000000000000000}=\frac{1003605945944011425233769242881280649744658171441 {\color{red}{\frac{t^{1 + 1}}{1 + 1}}}}{1000000000000000000000000000000000000000000000000}=\frac{1003605945944011425233769242881280649744658171441 {\color{red}{\left(\frac{t^{2}}{2}\right)}}}{1000000000000000000000000000000000000000000000000}$$
したがって、
$$\int{\frac{1003605945944011425233769242881280649744658171441 t}{1000000000000000000000000000000000000000000000000} d t} = \frac{1003605945944011425233769242881280649744658171441 t^{2}}{2000000000000000000000000000000000000000000000000}$$
積分定数を加える:
$$\int{\frac{1003605945944011425233769242881280649744658171441 t}{1000000000000000000000000000000000000000000000000} d t} = \frac{1003605945944011425233769242881280649744658171441 t^{2}}{2000000000000000000000000000000000000000000000000}+C$$
解答
$$$\int \frac{1003605945944011425233769242881280649744658171441 t}{1000000000000000000000000000000000000000000000000}\, dt = \frac{1003605945944011425233769242881280649744658171441 t^{2}}{2000000000000000000000000000000000000000000000000} + C$$$A