Integral of $$$212 x^{2} \sqrt{x^{31}}$$$

Find $$$\int 212 x^{2} \sqrt{x^{31}}\, dx$$$.

The input is rewritten: $$$\int{212 x^{2} \sqrt{x^{31}} d x}=\int{212 x^{\frac{35}{2}} d x}$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=212$$$ and $$$f{\left(x \right)} = x^{\frac{35}{2}}$$$:

$${\color{red}{\int{212 x^{\frac{35}{2}} d x}}} = {\color{red}{\left(212 \int{x^{\frac{35}{2}} d x}\right)}}$$

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=\frac{35}{2}$$$:

$$212 {\color{red}{\int{x^{\frac{35}{2}} d x}}}=212 {\color{red}{\frac{x^{1 + \frac{35}{2}}}{1 + \frac{35}{2}}}}=212 {\color{red}{\left(\frac{2 x^{\frac{37}{2}}}{37}\right)}}$$

Therefore,

$$\int{212 x^{\frac{35}{2}} d x} = \frac{424 x^{\frac{37}{2}}}{37}$$

Add the constant of integration:

$$\int{212 x^{\frac{35}{2}} d x} = \frac{424 x^{\frac{37}{2}}}{37}+C$$

$$$\int 212 x^{2} \sqrt{x^{31}}\, dx = \frac{424 x^{\frac{37}{2}}}{37} + C$$$A