Integral von $$$d_{} - j_{0} x^{2}$$$ nach $$$x$$$

Bestimme $$$\int \left(d_{} - j_{0} x^{2}\right)\, dx$$$.

Gliedweise integrieren:

$${\color{red}{\int{\left(d_{} - j_{0} x^{2}\right)d x}}} = {\color{red}{\left(\int{d_{} d x} - \int{j_{0} x^{2} d x}\right)}}$$

Wenden Sie die Konstantenregel $$$\int c\, dx = c x$$$ mit $$$c=d_{}$$$ an:

$$- \int{j_{0} x^{2} d x} + {\color{red}{\int{d_{} d x}}} = - \int{j_{0} x^{2} d x} + {\color{red}{d_{} x}}$$

Wende die Konstantenfaktorregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ mit $$$c=j_{0}$$$ und $$$f{\left(x \right)} = x^{2}$$$ an:

$$d_{} x - {\color{red}{\int{j_{0} x^{2} d x}}} = d_{} x - {\color{red}{j_{0} \int{x^{2} d x}}}$$

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

$$d_{} x - j_{0} {\color{red}{\int{x^{2} d x}}}=d_{} x - j_{0} {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=d_{} x - j_{0} {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

Daher,

$$\int{\left(d_{} - j_{0} x^{2}\right)d x} = d_{} x - \frac{j_{0} x^{3}}{3}$$

Vereinfachen:

$$\int{\left(d_{} - j_{0} x^{2}\right)d x} = x \left(d_{} - \frac{j_{0} x^{2}}{3}\right)$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{\left(d_{} - j_{0} x^{2}\right)d x} = x \left(d_{} - \frac{j_{0} x^{2}}{3}\right)+C$$

$$$\int \left(d_{} - j_{0} x^{2}\right)\, dx = x \left(d_{} - \frac{j_{0} x^{2}}{3}\right) + C$$$A