Integral of $$$x^{3} \sqrt{x - 1}$$$

Find $$$\int x^{3} \sqrt{x - 1}\, dx$$$.

Let $$$u=x - 1$$$.

Then $$$du=\left(x - 1\right)^{\prime }dx = 1 dx$$$ (steps can be seen »), and we have that $$$dx = du$$$.

The integral becomes

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

Expand the expression:

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

Integrate term by term:

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

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

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

Apply the power rule $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=\frac{7}{2}$$$:

$$\frac{2 u^{\frac{3}{2}}}{3} + \int{3 u^{\frac{3}{2}} d u} + \int{3 u^{\frac{5}{2}} d u} + {\color{red}{\int{u^{\frac{7}{2}} d u}}}=\frac{2 u^{\frac{3}{2}}}{3} + \int{3 u^{\frac{3}{2}} d u} + \int{3 u^{\frac{5}{2}} d u} + {\color{red}{\frac{u^{1 + \frac{7}{2}}}{1 + \frac{7}{2}}}}=\frac{2 u^{\frac{3}{2}}}{3} + \int{3 u^{\frac{3}{2}} d u} + \int{3 u^{\frac{5}{2}} d u} + {\color{red}{\left(\frac{2 u^{\frac{9}{2}}}{9}\right)}}$$

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

$$\frac{2 u^{\frac{9}{2}}}{9} + \frac{2 u^{\frac{3}{2}}}{3} + \int{3 u^{\frac{5}{2}} d u} + {\color{red}{\int{3 u^{\frac{3}{2}} d u}}} = \frac{2 u^{\frac{9}{2}}}{9} + \frac{2 u^{\frac{3}{2}}}{3} + \int{3 u^{\frac{5}{2}} d u} + {\color{red}{\left(3 \int{u^{\frac{3}{2}} d u}\right)}}$$

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

$$\frac{2 u^{\frac{9}{2}}}{9} + \frac{2 u^{\frac{3}{2}}}{3} + \int{3 u^{\frac{5}{2}} d u} + 3 {\color{red}{\int{u^{\frac{3}{2}} d u}}}=\frac{2 u^{\frac{9}{2}}}{9} + \frac{2 u^{\frac{3}{2}}}{3} + \int{3 u^{\frac{5}{2}} d u} + 3 {\color{red}{\frac{u^{1 + \frac{3}{2}}}{1 + \frac{3}{2}}}}=\frac{2 u^{\frac{9}{2}}}{9} + \frac{2 u^{\frac{3}{2}}}{3} + \int{3 u^{\frac{5}{2}} d u} + 3 {\color{red}{\left(\frac{2 u^{\frac{5}{2}}}{5}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=3$$$ and $$$f{\left(u \right)} = u^{\frac{5}{2}}$$$:

$$\frac{2 u^{\frac{9}{2}}}{9} + \frac{6 u^{\frac{5}{2}}}{5} + \frac{2 u^{\frac{3}{2}}}{3} + {\color{red}{\int{3 u^{\frac{5}{2}} d u}}} = \frac{2 u^{\frac{9}{2}}}{9} + \frac{6 u^{\frac{5}{2}}}{5} + \frac{2 u^{\frac{3}{2}}}{3} + {\color{red}{\left(3 \int{u^{\frac{5}{2}} d u}\right)}}$$

Apply the power rule $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=\frac{5}{2}$$$:

$$\frac{2 u^{\frac{9}{2}}}{9} + \frac{6 u^{\frac{5}{2}}}{5} + \frac{2 u^{\frac{3}{2}}}{3} + 3 {\color{red}{\int{u^{\frac{5}{2}} d u}}}=\frac{2 u^{\frac{9}{2}}}{9} + \frac{6 u^{\frac{5}{2}}}{5} + \frac{2 u^{\frac{3}{2}}}{3} + 3 {\color{red}{\frac{u^{1 + \frac{5}{2}}}{1 + \frac{5}{2}}}}=\frac{2 u^{\frac{9}{2}}}{9} + \frac{6 u^{\frac{5}{2}}}{5} + \frac{2 u^{\frac{3}{2}}}{3} + 3 {\color{red}{\left(\frac{2 u^{\frac{7}{2}}}{7}\right)}}$$

Recall that $$$u=x - 1$$$:

$$\frac{2 {\color{red}{u}}^{\frac{3}{2}}}{3} + \frac{6 {\color{red}{u}}^{\frac{5}{2}}}{5} + \frac{6 {\color{red}{u}}^{\frac{7}{2}}}{7} + \frac{2 {\color{red}{u}}^{\frac{9}{2}}}{9} = \frac{2 {\color{red}{\left(x - 1\right)}}^{\frac{3}{2}}}{3} + \frac{6 {\color{red}{\left(x - 1\right)}}^{\frac{5}{2}}}{5} + \frac{6 {\color{red}{\left(x - 1\right)}}^{\frac{7}{2}}}{7} + \frac{2 {\color{red}{\left(x - 1\right)}}^{\frac{9}{2}}}{9}$$

Therefore,

$$\int{x^{3} \sqrt{x - 1} d x} = \frac{2 \left(x - 1\right)^{\frac{9}{2}}}{9} + \frac{6 \left(x - 1\right)^{\frac{7}{2}}}{7} + \frac{6 \left(x - 1\right)^{\frac{5}{2}}}{5} + \frac{2 \left(x - 1\right)^{\frac{3}{2}}}{3}$$

Simplify:

$$\int{x^{3} \sqrt{x - 1} d x} = \frac{2 \left(x - 1\right)^{\frac{3}{2}} \left(35 x^{3} + 30 x^{2} + 24 x + 16\right)}{315}$$

Add the constant of integration:

$$\int{x^{3} \sqrt{x - 1} d x} = \frac{2 \left(x - 1\right)^{\frac{3}{2}} \left(35 x^{3} + 30 x^{2} + 24 x + 16\right)}{315}+C$$

$$$\int x^{3} \sqrt{x - 1}\, dx = \frac{2 \left(x - 1\right)^{\frac{3}{2}} \left(35 x^{3} + 30 x^{2} + 24 x + 16\right)}{315} + C$$$A