Integral of $$$- 3 x \sqrt{5 - x^{2}} - x + 7$$$

Find $$$\int \left(- 3 x \sqrt{5 - x^{2}} - x + 7\right)\, dx$$$.

Integrate term by term:

$${\color{red}{\int{\left(- 3 x \sqrt{5 - x^{2}} - x + 7\right)d x}}} = {\color{red}{\left(\int{7 d x} - \int{x d x} - \int{3 x \sqrt{5 - x^{2}} d x}\right)}}$$

Apply the constant rule $$$\int c\, dx = c x$$$ with $$$c=7$$$:

$$- \int{x d x} - \int{3 x \sqrt{5 - x^{2}} d x} + {\color{red}{\int{7 d x}}} = - \int{x d x} - \int{3 x \sqrt{5 - x^{2}} d x} + {\color{red}{\left(7 x\right)}}$$

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

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

Let $$$u=5 - x^{2}$$$.

Then $$$du=\left(5 - x^{2}\right)^{\prime }dx = - 2 x dx$$$ (steps can be seen »), and we have that $$$x dx = - \frac{du}{2}$$$.

The integral becomes

$$- \frac{x^{2}}{2} + 7 x - {\color{red}{\int{3 x \sqrt{5 - x^{2}} d x}}} = - \frac{x^{2}}{2} + 7 x - {\color{red}{\int{\left(- \frac{3 \sqrt{u}}{2}\right)d u}}}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=- \frac{3}{2}$$$ and $$$f{\left(u \right)} = \sqrt{u}$$$:

$$- \frac{x^{2}}{2} + 7 x - {\color{red}{\int{\left(- \frac{3 \sqrt{u}}{2}\right)d u}}} = - \frac{x^{2}}{2} + 7 x - {\color{red}{\left(- \frac{3 \int{\sqrt{u} d u}}{2}\right)}}$$

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

$$- \frac{x^{2}}{2} + 7 x + \frac{3 {\color{red}{\int{\sqrt{u} d u}}}}{2}=- \frac{x^{2}}{2} + 7 x + \frac{3 {\color{red}{\int{u^{\frac{1}{2}} d u}}}}{2}=- \frac{x^{2}}{2} + 7 x + \frac{3 {\color{red}{\frac{u^{\frac{1}{2} + 1}}{\frac{1}{2} + 1}}}}{2}=- \frac{x^{2}}{2} + 7 x + \frac{3 {\color{red}{\left(\frac{2 u^{\frac{3}{2}}}{3}\right)}}}{2}$$

Recall that $$$u=5 - x^{2}$$$:

$$- \frac{x^{2}}{2} + 7 x + {\color{red}{u}}^{\frac{3}{2}} = - \frac{x^{2}}{2} + 7 x + {\color{red}{\left(5 - x^{2}\right)}}^{\frac{3}{2}}$$

Therefore,

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

Add the constant of integration:

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

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