Integral von $$$- 4 \sqrt{30} x^{2} \sqrt{i n t} - x^{2} + 880$$$ nach $$$x$$$

Bestimme $$$\int \left(- 4 \sqrt{30} x^{2} \sqrt{i n t} - x^{2} + 880\right)\, dx$$$.

Die Eingabe wird umgeschrieben: $$$\int{\left(- 4 \sqrt{30} x^{2} \sqrt{i n t} - x^{2} + 880\right)d x}=\int{\left(- 4 \sqrt{30} \sqrt{i} \sqrt{n} \sqrt{t} x^{2} - x^{2} + 880\right)d x}$$$.

Gliedweise integrieren:

$${\color{red}{\int{\left(- 4 \sqrt{30} \sqrt{i} \sqrt{n} \sqrt{t} x^{2} - x^{2} + 880\right)d x}}} = {\color{red}{\left(\int{880 d x} - \int{x^{2} d x} - \int{4 \sqrt{30} \sqrt{i} \sqrt{n} \sqrt{t} x^{2} d x}\right)}}$$

Wenden Sie die Konstantenregel $$$\int c\, dx = c x$$$ mit $$$c=880$$$ an:

$$- \int{x^{2} d x} - \int{4 \sqrt{30} \sqrt{i} \sqrt{n} \sqrt{t} x^{2} d x} + {\color{red}{\int{880 d x}}} = - \int{x^{2} d x} - \int{4 \sqrt{30} \sqrt{i} \sqrt{n} \sqrt{t} x^{2} d x} + {\color{red}{\left(880 x\right)}}$$

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

$$880 x - \int{4 \sqrt{30} \sqrt{i} \sqrt{n} \sqrt{t} x^{2} d x} - {\color{red}{\int{x^{2} d x}}}=880 x - \int{4 \sqrt{30} \sqrt{i} \sqrt{n} \sqrt{t} x^{2} d x} - {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=880 x - \int{4 \sqrt{30} \sqrt{i} \sqrt{n} \sqrt{t} x^{2} d x} - {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

Wende die Konstantenfaktorregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ mit $$$c=4 \sqrt{30} \sqrt{i} \sqrt{n} \sqrt{t}$$$ und $$$f{\left(x \right)} = x^{2}$$$ an:

$$- \frac{x^{3}}{3} + 880 x - {\color{red}{\int{4 \sqrt{30} \sqrt{i} \sqrt{n} \sqrt{t} x^{2} d x}}} = - \frac{x^{3}}{3} + 880 x - {\color{red}{\left(4 \sqrt{30} \sqrt{i} \sqrt{n} \sqrt{t} \int{x^{2} d x}\right)}}$$

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

$$- 4 \sqrt{30} \sqrt{i} \sqrt{n} \sqrt{t} {\color{red}{\int{x^{2} d x}}} - \frac{x^{3}}{3} + 880 x=- 4 \sqrt{30} \sqrt{i} \sqrt{n} \sqrt{t} {\color{red}{\frac{x^{1 + 2}}{1 + 2}}} - \frac{x^{3}}{3} + 880 x=- 4 \sqrt{30} \sqrt{i} \sqrt{n} \sqrt{t} {\color{red}{\left(\frac{x^{3}}{3}\right)}} - \frac{x^{3}}{3} + 880 x$$

Daher,

$$\int{\left(- 4 \sqrt{30} \sqrt{i} \sqrt{n} \sqrt{t} x^{2} - x^{2} + 880\right)d x} = - \frac{4 \sqrt{30} \sqrt{i} \sqrt{n} \sqrt{t} x^{3}}{3} - \frac{x^{3}}{3} + 880 x$$

Vereinfachen:

$$\int{\left(- 4 \sqrt{30} \sqrt{i} \sqrt{n} \sqrt{t} x^{2} - x^{2} + 880\right)d x} = \frac{x \left(- 4 \sqrt{15} \sqrt{n} \sqrt{t} x^{2} \left(1 + i\right) - x^{2} + 2640\right)}{3}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{\left(- 4 \sqrt{30} \sqrt{i} \sqrt{n} \sqrt{t} x^{2} - x^{2} + 880\right)d x} = \frac{x \left(- 4 \sqrt{15} \sqrt{n} \sqrt{t} x^{2} \left(1 + i\right) - x^{2} + 2640\right)}{3}+C$$

$$$\int \left(- 4 \sqrt{30} x^{2} \sqrt{i n t} - x^{2} + 880\right)\, dx = \frac{x \left(- 4 \sqrt{15} \sqrt{n} \sqrt{t} x^{2} \left(1 + i\right) - x^{2} + 2640\right)}{3} + C$$$A