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$$$\frac{\left(x - 3\right)^{6}}{49 x^{32}}$$$の積分
入力内容
$$$\int \frac{\left(x - 3\right)^{6}}{49 x^{32}}\, dx$$$ を求めよ。
解答
定数倍の法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ を、$$$c=\frac{1}{49}$$$ と $$$f{\left(x \right)} = \frac{\left(x - 3\right)^{6}}{x^{32}}$$$ に対して適用する:
$${\color{red}{\int{\frac{\left(x - 3\right)^{6}}{49 x^{32}} d x}}} = {\color{red}{\left(\frac{\int{\frac{\left(x - 3\right)^{6}}{x^{32}} d x}}{49}\right)}}$$
Expand the expression:
$$\frac{{\color{red}{\int{\frac{\left(x - 3\right)^{6}}{x^{32}} d x}}}}{49} = \frac{{\color{red}{\int{\left(\frac{1}{x^{26}} - \frac{18}{x^{27}} + \frac{135}{x^{28}} - \frac{540}{x^{29}} + \frac{1215}{x^{30}} - \frac{1458}{x^{31}} + \frac{729}{x^{32}}\right)d x}}}}{49}$$
項別に積分せよ:
$$\frac{{\color{red}{\int{\left(\frac{1}{x^{26}} - \frac{18}{x^{27}} + \frac{135}{x^{28}} - \frac{540}{x^{29}} + \frac{1215}{x^{30}} - \frac{1458}{x^{31}} + \frac{729}{x^{32}}\right)d x}}}}{49} = \frac{{\color{red}{\left(\int{\frac{729}{x^{32}} d x} - \int{\frac{1458}{x^{31}} d x} + \int{\frac{1215}{x^{30}} d x} - \int{\frac{540}{x^{29}} d x} + \int{\frac{135}{x^{28}} d x} - \int{\frac{18}{x^{27}} d x} + \int{\frac{1}{x^{26}} d x}\right)}}}{49}$$
$$$n=-26$$$ を用いて、べき乗の法則 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ を適用します:
$$\frac{\int{\frac{729}{x^{32}} d x}}{49} - \frac{\int{\frac{1458}{x^{31}} d x}}{49} + \frac{\int{\frac{1215}{x^{30}} d x}}{49} - \frac{\int{\frac{540}{x^{29}} d x}}{49} + \frac{\int{\frac{135}{x^{28}} d x}}{49} - \frac{\int{\frac{18}{x^{27}} d x}}{49} + \frac{{\color{red}{\int{\frac{1}{x^{26}} d x}}}}{49}=\frac{\int{\frac{729}{x^{32}} d x}}{49} - \frac{\int{\frac{1458}{x^{31}} d x}}{49} + \frac{\int{\frac{1215}{x^{30}} d x}}{49} - \frac{\int{\frac{540}{x^{29}} d x}}{49} + \frac{\int{\frac{135}{x^{28}} d x}}{49} - \frac{\int{\frac{18}{x^{27}} d x}}{49} + \frac{{\color{red}{\int{x^{-26} d x}}}}{49}=\frac{\int{\frac{729}{x^{32}} d x}}{49} - \frac{\int{\frac{1458}{x^{31}} d x}}{49} + \frac{\int{\frac{1215}{x^{30}} d x}}{49} - \frac{\int{\frac{540}{x^{29}} d x}}{49} + \frac{\int{\frac{135}{x^{28}} d x}}{49} - \frac{\int{\frac{18}{x^{27}} d x}}{49} + \frac{{\color{red}{\frac{x^{-26 + 1}}{-26 + 1}}}}{49}=\frac{\int{\frac{729}{x^{32}} d x}}{49} - \frac{\int{\frac{1458}{x^{31}} d x}}{49} + \frac{\int{\frac{1215}{x^{30}} d x}}{49} - \frac{\int{\frac{540}{x^{29}} d x}}{49} + \frac{\int{\frac{135}{x^{28}} d x}}{49} - \frac{\int{\frac{18}{x^{27}} d x}}{49} + \frac{{\color{red}{\left(- \frac{x^{-25}}{25}\right)}}}{49}=\frac{\int{\frac{729}{x^{32}} d x}}{49} - \frac{\int{\frac{1458}{x^{31}} d x}}{49} + \frac{\int{\frac{1215}{x^{30}} d x}}{49} - \frac{\int{\frac{540}{x^{29}} d x}}{49} + \frac{\int{\frac{135}{x^{28}} d x}}{49} - \frac{\int{\frac{18}{x^{27}} d x}}{49} + \frac{{\color{red}{\left(- \frac{1}{25 x^{25}}\right)}}}{49}$$
定数倍の法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ を、$$$c=1458$$$ と $$$f{\left(x \right)} = \frac{1}{x^{31}}$$$ に対して適用する:
$$\frac{\int{\frac{729}{x^{32}} d x}}{49} + \frac{\int{\frac{1215}{x^{30}} d x}}{49} - \frac{\int{\frac{540}{x^{29}} d x}}{49} + \frac{\int{\frac{135}{x^{28}} d x}}{49} - \frac{\int{\frac{18}{x^{27}} d x}}{49} - \frac{{\color{red}{\int{\frac{1458}{x^{31}} d x}}}}{49} - \frac{1}{1225 x^{25}} = \frac{\int{\frac{729}{x^{32}} d x}}{49} + \frac{\int{\frac{1215}{x^{30}} d x}}{49} - \frac{\int{\frac{540}{x^{29}} d x}}{49} + \frac{\int{\frac{135}{x^{28}} d x}}{49} - \frac{\int{\frac{18}{x^{27}} d x}}{49} - \frac{{\color{red}{\left(1458 \int{\frac{1}{x^{31}} d x}\right)}}}{49} - \frac{1}{1225 x^{25}}$$
$$$n=-31$$$ を用いて、べき乗の法則 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ を適用します:
$$\frac{\int{\frac{729}{x^{32}} d x}}{49} + \frac{\int{\frac{1215}{x^{30}} d x}}{49} - \frac{\int{\frac{540}{x^{29}} d x}}{49} + \frac{\int{\frac{135}{x^{28}} d x}}{49} - \frac{\int{\frac{18}{x^{27}} d x}}{49} - \frac{1458 {\color{red}{\int{\frac{1}{x^{31}} d x}}}}{49} - \frac{1}{1225 x^{25}}=\frac{\int{\frac{729}{x^{32}} d x}}{49} + \frac{\int{\frac{1215}{x^{30}} d x}}{49} - \frac{\int{\frac{540}{x^{29}} d x}}{49} + \frac{\int{\frac{135}{x^{28}} d x}}{49} - \frac{\int{\frac{18}{x^{27}} d x}}{49} - \frac{1458 {\color{red}{\int{x^{-31} d x}}}}{49} - \frac{1}{1225 x^{25}}=\frac{\int{\frac{729}{x^{32}} d x}}{49} + \frac{\int{\frac{1215}{x^{30}} d x}}{49} - \frac{\int{\frac{540}{x^{29}} d x}}{49} + \frac{\int{\frac{135}{x^{28}} d x}}{49} - \frac{\int{\frac{18}{x^{27}} d x}}{49} - \frac{1458 {\color{red}{\frac{x^{-31 + 1}}{-31 + 1}}}}{49} - \frac{1}{1225 x^{25}}=\frac{\int{\frac{729}{x^{32}} d x}}{49} + \frac{\int{\frac{1215}{x^{30}} d x}}{49} - \frac{\int{\frac{540}{x^{29}} d x}}{49} + \frac{\int{\frac{135}{x^{28}} d x}}{49} - \frac{\int{\frac{18}{x^{27}} d x}}{49} - \frac{1458 {\color{red}{\left(- \frac{x^{-30}}{30}\right)}}}{49} - \frac{1}{1225 x^{25}}=\frac{\int{\frac{729}{x^{32}} d x}}{49} + \frac{\int{\frac{1215}{x^{30}} d x}}{49} - \frac{\int{\frac{540}{x^{29}} d x}}{49} + \frac{\int{\frac{135}{x^{28}} d x}}{49} - \frac{\int{\frac{18}{x^{27}} d x}}{49} - \frac{1458 {\color{red}{\left(- \frac{1}{30 x^{30}}\right)}}}{49} - \frac{1}{1225 x^{25}}$$
定数倍の法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ を、$$$c=540$$$ と $$$f{\left(x \right)} = \frac{1}{x^{29}}$$$ に対して適用する:
$$\frac{\int{\frac{729}{x^{32}} d x}}{49} + \frac{\int{\frac{1215}{x^{30}} d x}}{49} + \frac{\int{\frac{135}{x^{28}} d x}}{49} - \frac{\int{\frac{18}{x^{27}} d x}}{49} - \frac{{\color{red}{\int{\frac{540}{x^{29}} d x}}}}{49} - \frac{1}{1225 x^{25}} + \frac{243}{245 x^{30}} = \frac{\int{\frac{729}{x^{32}} d x}}{49} + \frac{\int{\frac{1215}{x^{30}} d x}}{49} + \frac{\int{\frac{135}{x^{28}} d x}}{49} - \frac{\int{\frac{18}{x^{27}} d x}}{49} - \frac{{\color{red}{\left(540 \int{\frac{1}{x^{29}} d x}\right)}}}{49} - \frac{1}{1225 x^{25}} + \frac{243}{245 x^{30}}$$
$$$n=-29$$$ を用いて、べき乗の法則 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ を適用します:
$$\frac{\int{\frac{729}{x^{32}} d x}}{49} + \frac{\int{\frac{1215}{x^{30}} d x}}{49} + \frac{\int{\frac{135}{x^{28}} d x}}{49} - \frac{\int{\frac{18}{x^{27}} d x}}{49} - \frac{540 {\color{red}{\int{\frac{1}{x^{29}} d x}}}}{49} - \frac{1}{1225 x^{25}} + \frac{243}{245 x^{30}}=\frac{\int{\frac{729}{x^{32}} d x}}{49} + \frac{\int{\frac{1215}{x^{30}} d x}}{49} + \frac{\int{\frac{135}{x^{28}} d x}}{49} - \frac{\int{\frac{18}{x^{27}} d x}}{49} - \frac{540 {\color{red}{\int{x^{-29} d x}}}}{49} - \frac{1}{1225 x^{25}} + \frac{243}{245 x^{30}}=\frac{\int{\frac{729}{x^{32}} d x}}{49} + \frac{\int{\frac{1215}{x^{30}} d x}}{49} + \frac{\int{\frac{135}{x^{28}} d x}}{49} - \frac{\int{\frac{18}{x^{27}} d x}}{49} - \frac{540 {\color{red}{\frac{x^{-29 + 1}}{-29 + 1}}}}{49} - \frac{1}{1225 x^{25}} + \frac{243}{245 x^{30}}=\frac{\int{\frac{729}{x^{32}} d x}}{49} + \frac{\int{\frac{1215}{x^{30}} d x}}{49} + \frac{\int{\frac{135}{x^{28}} d x}}{49} - \frac{\int{\frac{18}{x^{27}} d x}}{49} - \frac{540 {\color{red}{\left(- \frac{x^{-28}}{28}\right)}}}{49} - \frac{1}{1225 x^{25}} + \frac{243}{245 x^{30}}=\frac{\int{\frac{729}{x^{32}} d x}}{49} + \frac{\int{\frac{1215}{x^{30}} d x}}{49} + \frac{\int{\frac{135}{x^{28}} d x}}{49} - \frac{\int{\frac{18}{x^{27}} d x}}{49} - \frac{540 {\color{red}{\left(- \frac{1}{28 x^{28}}\right)}}}{49} - \frac{1}{1225 x^{25}} + \frac{243}{245 x^{30}}$$
定数倍の法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ を、$$$c=18$$$ と $$$f{\left(x \right)} = \frac{1}{x^{27}}$$$ に対して適用する:
$$\frac{\int{\frac{729}{x^{32}} d x}}{49} + \frac{\int{\frac{1215}{x^{30}} d x}}{49} + \frac{\int{\frac{135}{x^{28}} d x}}{49} - \frac{{\color{red}{\int{\frac{18}{x^{27}} d x}}}}{49} - \frac{1}{1225 x^{25}} + \frac{135}{343 x^{28}} + \frac{243}{245 x^{30}} = \frac{\int{\frac{729}{x^{32}} d x}}{49} + \frac{\int{\frac{1215}{x^{30}} d x}}{49} + \frac{\int{\frac{135}{x^{28}} d x}}{49} - \frac{{\color{red}{\left(18 \int{\frac{1}{x^{27}} d x}\right)}}}{49} - \frac{1}{1225 x^{25}} + \frac{135}{343 x^{28}} + \frac{243}{245 x^{30}}$$
$$$n=-27$$$ を用いて、べき乗の法則 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ を適用します:
$$\frac{\int{\frac{729}{x^{32}} d x}}{49} + \frac{\int{\frac{1215}{x^{30}} d x}}{49} + \frac{\int{\frac{135}{x^{28}} d x}}{49} - \frac{18 {\color{red}{\int{\frac{1}{x^{27}} d x}}}}{49} - \frac{1}{1225 x^{25}} + \frac{135}{343 x^{28}} + \frac{243}{245 x^{30}}=\frac{\int{\frac{729}{x^{32}} d x}}{49} + \frac{\int{\frac{1215}{x^{30}} d x}}{49} + \frac{\int{\frac{135}{x^{28}} d x}}{49} - \frac{18 {\color{red}{\int{x^{-27} d x}}}}{49} - \frac{1}{1225 x^{25}} + \frac{135}{343 x^{28}} + \frac{243}{245 x^{30}}=\frac{\int{\frac{729}{x^{32}} d x}}{49} + \frac{\int{\frac{1215}{x^{30}} d x}}{49} + \frac{\int{\frac{135}{x^{28}} d x}}{49} - \frac{18 {\color{red}{\frac{x^{-27 + 1}}{-27 + 1}}}}{49} - \frac{1}{1225 x^{25}} + \frac{135}{343 x^{28}} + \frac{243}{245 x^{30}}=\frac{\int{\frac{729}{x^{32}} d x}}{49} + \frac{\int{\frac{1215}{x^{30}} d x}}{49} + \frac{\int{\frac{135}{x^{28}} d x}}{49} - \frac{18 {\color{red}{\left(- \frac{x^{-26}}{26}\right)}}}{49} - \frac{1}{1225 x^{25}} + \frac{135}{343 x^{28}} + \frac{243}{245 x^{30}}=\frac{\int{\frac{729}{x^{32}} d x}}{49} + \frac{\int{\frac{1215}{x^{30}} d x}}{49} + \frac{\int{\frac{135}{x^{28}} d x}}{49} - \frac{18 {\color{red}{\left(- \frac{1}{26 x^{26}}\right)}}}{49} - \frac{1}{1225 x^{25}} + \frac{135}{343 x^{28}} + \frac{243}{245 x^{30}}$$
定数倍の法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ を、$$$c=135$$$ と $$$f{\left(x \right)} = \frac{1}{x^{28}}$$$ に対して適用する:
$$\frac{\int{\frac{729}{x^{32}} d x}}{49} + \frac{\int{\frac{1215}{x^{30}} d x}}{49} + \frac{{\color{red}{\int{\frac{135}{x^{28}} d x}}}}{49} - \frac{1}{1225 x^{25}} + \frac{9}{637 x^{26}} + \frac{135}{343 x^{28}} + \frac{243}{245 x^{30}} = \frac{\int{\frac{729}{x^{32}} d x}}{49} + \frac{\int{\frac{1215}{x^{30}} d x}}{49} + \frac{{\color{red}{\left(135 \int{\frac{1}{x^{28}} d x}\right)}}}{49} - \frac{1}{1225 x^{25}} + \frac{9}{637 x^{26}} + \frac{135}{343 x^{28}} + \frac{243}{245 x^{30}}$$
$$$n=-28$$$ を用いて、べき乗の法則 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ を適用します:
$$\frac{\int{\frac{729}{x^{32}} d x}}{49} + \frac{\int{\frac{1215}{x^{30}} d x}}{49} + \frac{135 {\color{red}{\int{\frac{1}{x^{28}} d x}}}}{49} - \frac{1}{1225 x^{25}} + \frac{9}{637 x^{26}} + \frac{135}{343 x^{28}} + \frac{243}{245 x^{30}}=\frac{\int{\frac{729}{x^{32}} d x}}{49} + \frac{\int{\frac{1215}{x^{30}} d x}}{49} + \frac{135 {\color{red}{\int{x^{-28} d x}}}}{49} - \frac{1}{1225 x^{25}} + \frac{9}{637 x^{26}} + \frac{135}{343 x^{28}} + \frac{243}{245 x^{30}}=\frac{\int{\frac{729}{x^{32}} d x}}{49} + \frac{\int{\frac{1215}{x^{30}} d x}}{49} + \frac{135 {\color{red}{\frac{x^{-28 + 1}}{-28 + 1}}}}{49} - \frac{1}{1225 x^{25}} + \frac{9}{637 x^{26}} + \frac{135}{343 x^{28}} + \frac{243}{245 x^{30}}=\frac{\int{\frac{729}{x^{32}} d x}}{49} + \frac{\int{\frac{1215}{x^{30}} d x}}{49} + \frac{135 {\color{red}{\left(- \frac{x^{-27}}{27}\right)}}}{49} - \frac{1}{1225 x^{25}} + \frac{9}{637 x^{26}} + \frac{135}{343 x^{28}} + \frac{243}{245 x^{30}}=\frac{\int{\frac{729}{x^{32}} d x}}{49} + \frac{\int{\frac{1215}{x^{30}} d x}}{49} + \frac{135 {\color{red}{\left(- \frac{1}{27 x^{27}}\right)}}}{49} - \frac{1}{1225 x^{25}} + \frac{9}{637 x^{26}} + \frac{135}{343 x^{28}} + \frac{243}{245 x^{30}}$$
定数倍の法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ を、$$$c=729$$$ と $$$f{\left(x \right)} = \frac{1}{x^{32}}$$$ に対して適用する:
$$\frac{\int{\frac{1215}{x^{30}} d x}}{49} + \frac{{\color{red}{\int{\frac{729}{x^{32}} d x}}}}{49} - \frac{1}{1225 x^{25}} + \frac{9}{637 x^{26}} - \frac{5}{49 x^{27}} + \frac{135}{343 x^{28}} + \frac{243}{245 x^{30}} = \frac{\int{\frac{1215}{x^{30}} d x}}{49} + \frac{{\color{red}{\left(729 \int{\frac{1}{x^{32}} d x}\right)}}}{49} - \frac{1}{1225 x^{25}} + \frac{9}{637 x^{26}} - \frac{5}{49 x^{27}} + \frac{135}{343 x^{28}} + \frac{243}{245 x^{30}}$$
$$$n=-32$$$ を用いて、べき乗の法則 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ を適用します:
$$\frac{\int{\frac{1215}{x^{30}} d x}}{49} + \frac{729 {\color{red}{\int{\frac{1}{x^{32}} d x}}}}{49} - \frac{1}{1225 x^{25}} + \frac{9}{637 x^{26}} - \frac{5}{49 x^{27}} + \frac{135}{343 x^{28}} + \frac{243}{245 x^{30}}=\frac{\int{\frac{1215}{x^{30}} d x}}{49} + \frac{729 {\color{red}{\int{x^{-32} d x}}}}{49} - \frac{1}{1225 x^{25}} + \frac{9}{637 x^{26}} - \frac{5}{49 x^{27}} + \frac{135}{343 x^{28}} + \frac{243}{245 x^{30}}=\frac{\int{\frac{1215}{x^{30}} d x}}{49} + \frac{729 {\color{red}{\frac{x^{-32 + 1}}{-32 + 1}}}}{49} - \frac{1}{1225 x^{25}} + \frac{9}{637 x^{26}} - \frac{5}{49 x^{27}} + \frac{135}{343 x^{28}} + \frac{243}{245 x^{30}}=\frac{\int{\frac{1215}{x^{30}} d x}}{49} + \frac{729 {\color{red}{\left(- \frac{x^{-31}}{31}\right)}}}{49} - \frac{1}{1225 x^{25}} + \frac{9}{637 x^{26}} - \frac{5}{49 x^{27}} + \frac{135}{343 x^{28}} + \frac{243}{245 x^{30}}=\frac{\int{\frac{1215}{x^{30}} d x}}{49} + \frac{729 {\color{red}{\left(- \frac{1}{31 x^{31}}\right)}}}{49} - \frac{1}{1225 x^{25}} + \frac{9}{637 x^{26}} - \frac{5}{49 x^{27}} + \frac{135}{343 x^{28}} + \frac{243}{245 x^{30}}$$
定数倍の法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ を、$$$c=1215$$$ と $$$f{\left(x \right)} = \frac{1}{x^{30}}$$$ に対して適用する:
$$\frac{{\color{red}{\int{\frac{1215}{x^{30}} d x}}}}{49} - \frac{1}{1225 x^{25}} + \frac{9}{637 x^{26}} - \frac{5}{49 x^{27}} + \frac{135}{343 x^{28}} + \frac{243}{245 x^{30}} - \frac{729}{1519 x^{31}} = \frac{{\color{red}{\left(1215 \int{\frac{1}{x^{30}} d x}\right)}}}{49} - \frac{1}{1225 x^{25}} + \frac{9}{637 x^{26}} - \frac{5}{49 x^{27}} + \frac{135}{343 x^{28}} + \frac{243}{245 x^{30}} - \frac{729}{1519 x^{31}}$$
$$$n=-30$$$ を用いて、べき乗の法則 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ を適用します:
$$\frac{1215 {\color{red}{\int{\frac{1}{x^{30}} d x}}}}{49} - \frac{1}{1225 x^{25}} + \frac{9}{637 x^{26}} - \frac{5}{49 x^{27}} + \frac{135}{343 x^{28}} + \frac{243}{245 x^{30}} - \frac{729}{1519 x^{31}}=\frac{1215 {\color{red}{\int{x^{-30} d x}}}}{49} - \frac{1}{1225 x^{25}} + \frac{9}{637 x^{26}} - \frac{5}{49 x^{27}} + \frac{135}{343 x^{28}} + \frac{243}{245 x^{30}} - \frac{729}{1519 x^{31}}=\frac{1215 {\color{red}{\frac{x^{-30 + 1}}{-30 + 1}}}}{49} - \frac{1}{1225 x^{25}} + \frac{9}{637 x^{26}} - \frac{5}{49 x^{27}} + \frac{135}{343 x^{28}} + \frac{243}{245 x^{30}} - \frac{729}{1519 x^{31}}=\frac{1215 {\color{red}{\left(- \frac{x^{-29}}{29}\right)}}}{49} - \frac{1}{1225 x^{25}} + \frac{9}{637 x^{26}} - \frac{5}{49 x^{27}} + \frac{135}{343 x^{28}} + \frac{243}{245 x^{30}} - \frac{729}{1519 x^{31}}=\frac{1215 {\color{red}{\left(- \frac{1}{29 x^{29}}\right)}}}{49} - \frac{1}{1225 x^{25}} + \frac{9}{637 x^{26}} - \frac{5}{49 x^{27}} + \frac{135}{343 x^{28}} + \frac{243}{245 x^{30}} - \frac{729}{1519 x^{31}}$$
したがって、
$$\int{\frac{\left(x - 3\right)^{6}}{49 x^{32}} d x} = - \frac{1}{1225 x^{25}} + \frac{9}{637 x^{26}} - \frac{5}{49 x^{27}} + \frac{135}{343 x^{28}} - \frac{1215}{1421 x^{29}} + \frac{243}{245 x^{30}} - \frac{729}{1519 x^{31}}$$
簡単化せよ:
$$\int{\frac{\left(x - 3\right)^{6}}{49 x^{32}} d x} = \frac{- 81809 x^{6} + 1415925 x^{5} - 10226125 x^{4} + 39443625 x^{3} - 85687875 x^{2} + 99397935 x - 48095775}{100216025 x^{31}}$$
積分定数を加える:
$$\int{\frac{\left(x - 3\right)^{6}}{49 x^{32}} d x} = \frac{- 81809 x^{6} + 1415925 x^{5} - 10226125 x^{4} + 39443625 x^{3} - 85687875 x^{2} + 99397935 x - 48095775}{100216025 x^{31}}+C$$
解答
$$$\int \frac{\left(x - 3\right)^{6}}{49 x^{32}}\, dx = \frac{- 81809 x^{6} + 1415925 x^{5} - 10226125 x^{4} + 39443625 x^{3} - 85687875 x^{2} + 99397935 x - 48095775}{100216025 x^{31}} + C$$$A