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

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

Integrate term by term:

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

Apply the constant rule $$$\int c\, dx = c x$$$ with $$$c=15625$$$:

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

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=3$$$ and $$$f{\left(x \right)} = x^{2}$$$:

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

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=2$$$:

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

Therefore,

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

Simplify:

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

Add the constant of integration:

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

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