Integral of $$$- \frac{6}{\sqrt{y^{3}}} + \frac{3}{\sqrt{y}}$$$

Find $$$\int \left(- \frac{6}{\sqrt{y^{3}}} + \frac{3}{\sqrt{y}}\right)\, dy$$$.

The input is rewritten: $$$\int{\left(- \frac{6}{\sqrt{y^{3}}} + \frac{3}{\sqrt{y}}\right)d y}=\int{\left(\frac{3}{\sqrt{y}} - \frac{6}{y^{\frac{3}{2}}}\right)d y}$$$.

Integrate term by term:

$${\color{red}{\int{\left(\frac{3}{\sqrt{y}} - \frac{6}{y^{\frac{3}{2}}}\right)d y}}} = {\color{red}{\left(- \int{\frac{6}{y^{\frac{3}{2}}} d y} + \int{\frac{3}{\sqrt{y}} d y}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(y \right)}\, dy = c \int f{\left(y \right)}\, dy$$$ with $$$c=6$$$ and $$$f{\left(y \right)} = \frac{1}{y^{\frac{3}{2}}}$$$:

$$\int{\frac{3}{\sqrt{y}} d y} - {\color{red}{\int{\frac{6}{y^{\frac{3}{2}}} d y}}} = \int{\frac{3}{\sqrt{y}} d y} - {\color{red}{\left(6 \int{\frac{1}{y^{\frac{3}{2}}} d y}\right)}}$$

Apply the power rule $$$\int y^{n}\, dy = \frac{y^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=- \frac{3}{2}$$$:

$$\int{\frac{3}{\sqrt{y}} d y} - 6 {\color{red}{\int{\frac{1}{y^{\frac{3}{2}}} d y}}}=\int{\frac{3}{\sqrt{y}} d y} - 6 {\color{red}{\int{y^{- \frac{3}{2}} d y}}}=\int{\frac{3}{\sqrt{y}} d y} - 6 {\color{red}{\frac{y^{- \frac{3}{2} + 1}}{- \frac{3}{2} + 1}}}=\int{\frac{3}{\sqrt{y}} d y} - 6 {\color{red}{\left(- 2 y^{- \frac{1}{2}}\right)}}=\int{\frac{3}{\sqrt{y}} d y} - 6 {\color{red}{\left(- \frac{2}{\sqrt{y}}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(y \right)}\, dy = c \int f{\left(y \right)}\, dy$$$ with $$$c=3$$$ and $$$f{\left(y \right)} = \frac{1}{\sqrt{y}}$$$:

$${\color{red}{\int{\frac{3}{\sqrt{y}} d y}}} + \frac{12}{\sqrt{y}} = {\color{red}{\left(3 \int{\frac{1}{\sqrt{y}} d y}\right)}} + \frac{12}{\sqrt{y}}$$

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

$$3 {\color{red}{\int{\frac{1}{\sqrt{y}} d y}}} + \frac{12}{\sqrt{y}}=3 {\color{red}{\int{y^{- \frac{1}{2}} d y}}} + \frac{12}{\sqrt{y}}=3 {\color{red}{\frac{y^{- \frac{1}{2} + 1}}{- \frac{1}{2} + 1}}} + \frac{12}{\sqrt{y}}=3 {\color{red}{\left(2 y^{\frac{1}{2}}\right)}} + \frac{12}{\sqrt{y}}=3 {\color{red}{\left(2 \sqrt{y}\right)}} + \frac{12}{\sqrt{y}}$$

Therefore,

$$\int{\left(\frac{3}{\sqrt{y}} - \frac{6}{y^{\frac{3}{2}}}\right)d y} = 6 \sqrt{y} + \frac{12}{\sqrt{y}}$$

Simplify:

$$\int{\left(\frac{3}{\sqrt{y}} - \frac{6}{y^{\frac{3}{2}}}\right)d y} = \frac{6 \left(y + 2\right)}{\sqrt{y}}$$

Add the constant of integration:

$$\int{\left(\frac{3}{\sqrt{y}} - \frac{6}{y^{\frac{3}{2}}}\right)d y} = \frac{6 \left(y + 2\right)}{\sqrt{y}}+C$$

$$$\int \left(- \frac{6}{\sqrt{y^{3}}} + \frac{3}{\sqrt{y}}\right)\, dy = \frac{6 \left(y + 2\right)}{\sqrt{y}} + C$$$A