Integral of $$$- x^{2} + \frac{\sqrt{10}}{10 \sqrt{x}}$$$

Find $$$\int \left(- x^{2} + \frac{\sqrt{10}}{10 \sqrt{x}}\right)\, dx$$$.

Integrate term by term:

$${\color{red}{\int{\left(- x^{2} + \frac{\sqrt{10}}{10 \sqrt{x}}\right)d x}}} = {\color{red}{\left(- \int{x^{2} d x} + \int{\frac{\sqrt{10}}{10 \sqrt{x}} d x}\right)}}$$

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

$$\int{\frac{\sqrt{10}}{10 \sqrt{x}} d x} - {\color{red}{\int{x^{2} d x}}}=\int{\frac{\sqrt{10}}{10 \sqrt{x}} d x} - {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=\int{\frac{\sqrt{10}}{10 \sqrt{x}} d x} - {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=\frac{\sqrt{10}}{10}$$$ and $$$f{\left(x \right)} = \frac{1}{\sqrt{x}}$$$:

$$- \frac{x^{3}}{3} + {\color{red}{\int{\frac{\sqrt{10}}{10 \sqrt{x}} d x}}} = - \frac{x^{3}}{3} + {\color{red}{\left(\frac{\sqrt{10} \int{\frac{1}{\sqrt{x}} d x}}{10}\right)}}$$

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

$$- \frac{x^{3}}{3} + \frac{\sqrt{10} {\color{red}{\int{\frac{1}{\sqrt{x}} d x}}}}{10}=- \frac{x^{3}}{3} + \frac{\sqrt{10} {\color{red}{\int{x^{- \frac{1}{2}} d x}}}}{10}=- \frac{x^{3}}{3} + \frac{\sqrt{10} {\color{red}{\frac{x^{- \frac{1}{2} + 1}}{- \frac{1}{2} + 1}}}}{10}=- \frac{x^{3}}{3} + \frac{\sqrt{10} {\color{red}{\left(2 x^{\frac{1}{2}}\right)}}}{10}=- \frac{x^{3}}{3} + \frac{\sqrt{10} {\color{red}{\left(2 \sqrt{x}\right)}}}{10}$$

Therefore,

$$\int{\left(- x^{2} + \frac{\sqrt{10}}{10 \sqrt{x}}\right)d x} = \frac{\sqrt{10} \sqrt{x}}{5} - \frac{x^{3}}{3}$$

Add the constant of integration:

$$\int{\left(- x^{2} + \frac{\sqrt{10}}{10 \sqrt{x}}\right)d x} = \frac{\sqrt{10} \sqrt{x}}{5} - \frac{x^{3}}{3}+C$$

$$$\int \left(- x^{2} + \frac{\sqrt{10}}{10 \sqrt{x}}\right)\, dx = \left(\frac{\sqrt{10} \sqrt{x}}{5} - \frac{x^{3}}{3}\right) + C$$$A