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$$$\frac{13}{x^{\frac{28}{5}}}$$$ 的積分
您的輸入
求$$$\int \frac{13}{x^{\frac{28}{5}}}\, dx$$$。
解答
套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=13$$$ 與 $$$f{\left(x \right)} = \frac{1}{x^{\frac{28}{5}}}$$$：
$${\color{red}{\int{\frac{13}{x^{\frac{28}{5}}} d x}}} = {\color{red}{\left(13 \int{\frac{1}{x^{\frac{28}{5}}} d x}\right)}}$$
套用冪次法則 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，以 $$$n=- \frac{28}{5}$$$：
$$13 {\color{red}{\int{\frac{1}{x^{\frac{28}{5}}} d x}}}=13 {\color{red}{\int{x^{- \frac{28}{5}} d x}}}=13 {\color{red}{\frac{x^{- \frac{28}{5} + 1}}{- \frac{28}{5} + 1}}}=13 {\color{red}{\left(- \frac{5 x^{- \frac{23}{5}}}{23}\right)}}=13 {\color{red}{\left(- \frac{5}{23 x^{\frac{23}{5}}}\right)}}$$
因此，
$$\int{\frac{13}{x^{\frac{28}{5}}} d x} = - \frac{65}{23 x^{\frac{23}{5}}}$$
加上積分常數：
$$\int{\frac{13}{x^{\frac{28}{5}}} d x} = - \frac{65}{23 x^{\frac{23}{5}}}+C$$
答案
$$$\int \frac{13}{x^{\frac{28}{5}}}\, dx = - \frac{65}{23 x^{\frac{23}{5}}} + C$$$A