∫〖2x+1〗

$$\int 2x+1\;dx = \frac{2x^2}{2}+x+c\;=x^2+x+c$$

it = 2 okey 

∫〖2x+1〗  ?

$$\int{(2x+1)\ dx } \\\\ = \int{2x\ dx } + \int{1\ dx }\quad | \quad 1=x^0 \;!\\\\ = 2\int{x^{\textcolor[rgb]{1,0,0}1} \ dx } + \int{x^{\textcolor[rgb]{1,0,0}0} \ dx }$$

$$\boxed { \int{x^{\textcolor[rgb]{1,0,0}{n}} \ dx } = { x^{\textcolor[rgb]{1,0,0}{n}+1} \over \textcolor[rgb]{1,0,0}{n}+1} }$$

$$\\= 2 * \left( \dfrac{ x^{\textcolor[rgb]{1,0,0}{1} +1} }{ \textcolor[rgb]{1,0,0}{1} +1 } \right)+ \dfrac{ x^{\textcolor[rgb]{1,0,0}{0} +1} }{ \textcolor[rgb]{1,0,0}{0} +1 } + c\\\\ =\not{2} \dfrac{ x^{2} }{ \not{2}} + \dfrac{ x^{1} }{1 } + c\\\\ = x^{2}+x + c$$