Integral von $$$\frac{\sqrt{7}}{7 \sqrt{y}}$$$

Bestimme $$$\int \frac{\sqrt{7}}{7 \sqrt{y}}\, dy$$$.

Wende die Konstantenfaktorregel $$$\int c f{\left(y \right)}\, dy = c \int f{\left(y \right)}\, dy$$$ mit $$$c=\frac{\sqrt{7}}{7}$$$ und $$$f{\left(y \right)} = \frac{1}{\sqrt{y}}$$$ an:

$${\color{red}{\int{\frac{\sqrt{7}}{7 \sqrt{y}} d y}}} = {\color{red}{\left(\frac{\sqrt{7} \int{\frac{1}{\sqrt{y}} d y}}{7}\right)}}$$

Wenden Sie die Potenzregel $$$\int y^{n}\, dy = \frac{y^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ mit $$$n=- \frac{1}{2}$$$ an:

$$\frac{\sqrt{7} {\color{red}{\int{\frac{1}{\sqrt{y}} d y}}}}{7}=\frac{\sqrt{7} {\color{red}{\int{y^{- \frac{1}{2}} d y}}}}{7}=\frac{\sqrt{7} {\color{red}{\frac{y^{- \frac{1}{2} + 1}}{- \frac{1}{2} + 1}}}}{7}=\frac{\sqrt{7} {\color{red}{\left(2 y^{\frac{1}{2}}\right)}}}{7}=\frac{\sqrt{7} {\color{red}{\left(2 \sqrt{y}\right)}}}{7}$$

Daher,

$$\int{\frac{\sqrt{7}}{7 \sqrt{y}} d y} = \frac{2 \sqrt{7} \sqrt{y}}{7}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{\frac{\sqrt{7}}{7 \sqrt{y}} d y} = \frac{2 \sqrt{7} \sqrt{y}}{7}+C$$

$$$\int \frac{\sqrt{7}}{7 \sqrt{y}}\, dy = \frac{2 \sqrt{7} \sqrt{y}}{7} + C$$$A