Integral von $$$\frac{i d n t}{x^{115}}$$$ nach $$$x$$$

Bestimme $$$\int \frac{i d n t}{x^{115}}\, dx$$$.

Wende die Konstantenfaktorregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ mit $$$c=i d n t$$$ und $$$f{\left(x \right)} = \frac{1}{x^{115}}$$$ an:

$${\color{red}{\int{\frac{i d n t}{x^{115}} d x}}} = {\color{red}{i d n t \int{\frac{1}{x^{115}} d x}}}$$

Wenden Sie die Potenzregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ mit $$$n=-115$$$ an:

$$i d n t {\color{red}{\int{\frac{1}{x^{115}} d x}}}=i d n t {\color{red}{\int{x^{-115} d x}}}=i d n t {\color{red}{\frac{x^{-115 + 1}}{-115 + 1}}}=i d n t {\color{red}{\left(- \frac{x^{-114}}{114}\right)}}=i d n t {\color{red}{\left(- \frac{1}{114 x^{114}}\right)}}$$

Daher,

$$\int{\frac{i d n t}{x^{115}} d x} = - \frac{i d n t}{114 x^{114}}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{\frac{i d n t}{x^{115}} d x} = - \frac{i d n t}{114 x^{114}}+C$$

$$$\int \frac{i d n t}{x^{115}}\, dx = - \frac{i d n t}{114 x^{114}} + C$$$A