Integral of $$$5 x^{4} - 2 \sin{\left(x \right)} + \frac{1}{x^{3}}$$$

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

Integrate term by term:

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

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

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

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

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

The integral of the sine is $$$\int{\sin{\left(x \right)} d x} = - \cos{\left(x \right)}$$$:

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

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=5$$$ and $$$f{\left(x \right)} = x^{4}$$$:

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

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=4$$$:

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

Therefore,

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

Add the constant of integration:

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

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