Integral of $$$x \sqrt{4 - 7 x}$$$

Find $$$\int x \sqrt{4 - 7 x}\, dx$$$.

Let $$$u=4 - 7 x$$$.

Then $$$du=\left(4 - 7 x\right)^{\prime }dx = - 7 dx$$$ (steps can be seen »), and we have that $$$dx = - \frac{du}{7}$$$.

Thus,

$${\color{red}{\int{x \sqrt{4 - 7 x} d x}}} = {\color{red}{\int{\frac{\sqrt{u} \left(u - 4\right)}{49} d u}}}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=\frac{1}{49}$$$ and $$$f{\left(u \right)} = \sqrt{u} \left(u - 4\right)$$$:

$${\color{red}{\int{\frac{\sqrt{u} \left(u - 4\right)}{49} d u}}} = {\color{red}{\left(\frac{\int{\sqrt{u} \left(u - 4\right) d u}}{49}\right)}}$$

Expand the expression:

$$\frac{{\color{red}{\int{\sqrt{u} \left(u - 4\right) d u}}}}{49} = \frac{{\color{red}{\int{\left(u^{\frac{3}{2}} - 4 \sqrt{u}\right)d u}}}}{49}$$

Integrate term by term:

$$\frac{{\color{red}{\int{\left(u^{\frac{3}{2}} - 4 \sqrt{u}\right)d u}}}}{49} = \frac{{\color{red}{\left(- \int{4 \sqrt{u} d u} + \int{u^{\frac{3}{2}} d u}\right)}}}{49}$$

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

$$- \frac{\int{4 \sqrt{u} d u}}{49} + \frac{{\color{red}{\int{u^{\frac{3}{2}} d u}}}}{49}=- \frac{\int{4 \sqrt{u} d u}}{49} + \frac{{\color{red}{\frac{u^{1 + \frac{3}{2}}}{1 + \frac{3}{2}}}}}{49}=- \frac{\int{4 \sqrt{u} d u}}{49} + \frac{{\color{red}{\left(\frac{2 u^{\frac{5}{2}}}{5}\right)}}}{49}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=4$$$ and $$$f{\left(u \right)} = \sqrt{u}$$$:

$$\frac{2 u^{\frac{5}{2}}}{245} - \frac{{\color{red}{\int{4 \sqrt{u} d u}}}}{49} = \frac{2 u^{\frac{5}{2}}}{245} - \frac{{\color{red}{\left(4 \int{\sqrt{u} d u}\right)}}}{49}$$

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

$$\frac{2 u^{\frac{5}{2}}}{245} - \frac{4 {\color{red}{\int{\sqrt{u} d u}}}}{49}=\frac{2 u^{\frac{5}{2}}}{245} - \frac{4 {\color{red}{\int{u^{\frac{1}{2}} d u}}}}{49}=\frac{2 u^{\frac{5}{2}}}{245} - \frac{4 {\color{red}{\frac{u^{\frac{1}{2} + 1}}{\frac{1}{2} + 1}}}}{49}=\frac{2 u^{\frac{5}{2}}}{245} - \frac{4 {\color{red}{\left(\frac{2 u^{\frac{3}{2}}}{3}\right)}}}{49}$$

Recall that $$$u=4 - 7 x$$$:

$$- \frac{8 {\color{red}{u}}^{\frac{3}{2}}}{147} + \frac{2 {\color{red}{u}}^{\frac{5}{2}}}{245} = - \frac{8 {\color{red}{\left(4 - 7 x\right)}}^{\frac{3}{2}}}{147} + \frac{2 {\color{red}{\left(4 - 7 x\right)}}^{\frac{5}{2}}}{245}$$

Therefore,

$$\int{x \sqrt{4 - 7 x} d x} = \frac{2 \left(4 - 7 x\right)^{\frac{5}{2}}}{245} - \frac{8 \left(4 - 7 x\right)^{\frac{3}{2}}}{147}$$

Simplify:

$$\int{x \sqrt{4 - 7 x} d x} = \frac{2 \left(4 - 7 x\right)^{\frac{3}{2}} \left(- 21 x - 8\right)}{735}$$

Add the constant of integration:

$$\int{x \sqrt{4 - 7 x} d x} = \frac{2 \left(4 - 7 x\right)^{\frac{3}{2}} \left(- 21 x - 8\right)}{735}+C$$

$$$\int x \sqrt{4 - 7 x}\, dx = \frac{2 \left(4 - 7 x\right)^{\frac{3}{2}} \left(- 21 x - 8\right)}{735} + C$$$A