Integral von $$$3 x^{2} - 15625$$$

Bestimme $$$\int \left(3 x^{2} - 15625\right)\, dx$$$.

Gliedweise integrieren:

$${\color{red}{\int{\left(3 x^{2} - 15625\right)d x}}} = {\color{red}{\left(- \int{15625 d x} + \int{3 x^{2} d x}\right)}}$$

Wenden Sie die Konstantenregel $$$\int c\, dx = c x$$$ mit $$$c=15625$$$ an:

$$\int{3 x^{2} d x} - {\color{red}{\int{15625 d x}}} = \int{3 x^{2} d x} - {\color{red}{\left(15625 x\right)}}$$

Wende die Konstantenfaktorregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ mit $$$c=3$$$ und $$$f{\left(x \right)} = x^{2}$$$ an:

$$- 15625 x + {\color{red}{\int{3 x^{2} d x}}} = - 15625 x + {\color{red}{\left(3 \int{x^{2} d x}\right)}}$$

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

$$- 15625 x + 3 {\color{red}{\int{x^{2} d x}}}=- 15625 x + 3 {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=- 15625 x + 3 {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

Daher,

$$\int{\left(3 x^{2} - 15625\right)d x} = x^{3} - 15625 x$$

Vereinfachen:

$$\int{\left(3 x^{2} - 15625\right)d x} = x \left(x^{2} - 15625\right)$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{\left(3 x^{2} - 15625\right)d x} = x \left(x^{2} - 15625\right)+C$$

$$$\int \left(3 x^{2} - 15625\right)\, dx = x \left(x^{2} - 15625\right) + C$$$A