Integral of $$$\sqrt{2} x^{\frac{5}{2}} - 4 x^{3} - 1$$$

Find $$$\int \left(\sqrt{2} x^{\frac{5}{2}} - 4 x^{3} - 1\right)\, dx$$$.

Integrate term by term:

$${\color{red}{\int{\left(\sqrt{2} x^{\frac{5}{2}} - 4 x^{3} - 1\right)d x}}} = {\color{red}{\left(- \int{1 d x} - \int{4 x^{3} d x} + \int{\sqrt{2} x^{\frac{5}{2}} d x}\right)}}$$

Apply the constant rule $$$\int c\, dx = c x$$$ with $$$c=1$$$:

$$- \int{4 x^{3} d x} + \int{\sqrt{2} x^{\frac{5}{2}} d x} - {\color{red}{\int{1 d x}}} = - \int{4 x^{3} d x} + \int{\sqrt{2} x^{\frac{5}{2}} d x} - {\color{red}{x}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=4$$$ and $$$f{\left(x \right)} = x^{3}$$$:

$$- x + \int{\sqrt{2} x^{\frac{5}{2}} d x} - {\color{red}{\int{4 x^{3} d x}}} = - x + \int{\sqrt{2} x^{\frac{5}{2}} d x} - {\color{red}{\left(4 \int{x^{3} d x}\right)}}$$

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=3$$$:

$$- x + \int{\sqrt{2} x^{\frac{5}{2}} d x} - 4 {\color{red}{\int{x^{3} d x}}}=- x + \int{\sqrt{2} x^{\frac{5}{2}} d x} - 4 {\color{red}{\frac{x^{1 + 3}}{1 + 3}}}=- x + \int{\sqrt{2} x^{\frac{5}{2}} d x} - 4 {\color{red}{\left(\frac{x^{4}}{4}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=\sqrt{2}$$$ and $$$f{\left(x \right)} = x^{\frac{5}{2}}$$$:

$$- x^{4} - x + {\color{red}{\int{\sqrt{2} x^{\frac{5}{2}} d x}}} = - x^{4} - x + {\color{red}{\sqrt{2} \int{x^{\frac{5}{2}} d x}}}$$

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=\frac{5}{2}$$$:

$$- x^{4} - x + \sqrt{2} {\color{red}{\int{x^{\frac{5}{2}} d x}}}=- x^{4} - x + \sqrt{2} {\color{red}{\frac{x^{1 + \frac{5}{2}}}{1 + \frac{5}{2}}}}=- x^{4} - x + \sqrt{2} {\color{red}{\left(\frac{2 x^{\frac{7}{2}}}{7}\right)}}$$

Therefore,

$$\int{\left(\sqrt{2} x^{\frac{5}{2}} - 4 x^{3} - 1\right)d x} = \frac{2 \sqrt{2} x^{\frac{7}{2}}}{7} - x^{4} - x$$

Add the constant of integration:

$$\int{\left(\sqrt{2} x^{\frac{5}{2}} - 4 x^{3} - 1\right)d x} = \frac{2 \sqrt{2} x^{\frac{7}{2}}}{7} - x^{4} - x+C$$

$$$\int \left(\sqrt{2} x^{\frac{5}{2}} - 4 x^{3} - 1\right)\, dx = \left(\frac{2 \sqrt{2} x^{\frac{7}{2}}}{7} - x^{4} - x\right) + C$$$A