Evaluate: integral of (sqrt(x)+x+x^2)/(2x) x

Expression: $\int{ \frac{ \sqrt{ x }+x+{x}^{2} }{ 2x } } \mathrm{d} x$

Use the property of integral $\begin{array} { l }\int{ a \times f\left( x \right) } \mathrm{d} x=a \times \int{ f\left( x \right) } \mathrm{d} x,& a \in ℝ\end{array}$

$\frac{ 1 }{ 2 } \times \int{ \frac{ \sqrt{ x }+x+{x}^{2} }{ x } } \mathrm{d} x$

Use $\sqrt[n]{{a}^{m}}={a}^{\frac{ m }{ n }}$ to transform the expression

$\frac{ 1 }{ 2 } \times \int{ \frac{ {x}^{\frac{ 1 }{ 2 }}+x+{x}^{2} }{ x } } \mathrm{d} x$

Separate the fraction into $3$ fractions

$\frac{ 1 }{ 2 } \times \int{ \frac{ {x}^{\frac{ 1 }{ 2 }} }{ x }+\frac{ x }{ x }+\frac{ {x}^{2} }{ x } } \mathrm{d} x$

Simplify the expression

$\frac{ 1 }{ 2 } \times \int{ \frac{ 1 }{ {x}^{\frac{ 1 }{ 2 }} }+\frac{ x }{ x }+\frac{ {x}^{2} }{ x } } \mathrm{d} x$

Any expression divided by itself equals $1$

$\frac{ 1 }{ 2 } \times \int{ \frac{ 1 }{ {x}^{\frac{ 1 }{ 2 }} }+1+\frac{ {x}^{2} }{ x } } \mathrm{d} x$

Cancel out the common factor $x$

$\frac{ 1 }{ 2 } \times \int{ \frac{ 1 }{ {x}^{\frac{ 1 }{ 2 }} }+1+x } \mathrm{d} x$

Use the property of integral $\int{ f\left( x \right)\pmg\left( x \right) } \mathrm{d} x=\int{ f\left( x \right) } \mathrm{d} x\pm\int{ g\left( x \right) } \mathrm{d} x$

$\frac{ 1 }{ 2 } \times \left( \int{ \frac{ 1 }{ {x}^{\frac{ 1 }{ 2 }} } } \mathrm{d} x+\int{ 1 } \mathrm{d} x+\int{ x } \mathrm{d} x \right)$

Use $\begin{array} { l }\int{ \frac{ 1 }{ {x}^{n} } } \mathrm{d} x=-\frac{ 1 }{ \left( n-1 \right) \times {x}^{n-1} },& n≠1\end{array}$ to evaluate the integral

$\frac{ 1 }{ 2 } \times \left( 2\sqrt{ x }+\int{ 1 } \mathrm{d} x+\int{ x } \mathrm{d} x \right)$

Use $\int{ 1 } \mathrm{d} x=x$ to evaluate the integral

$\frac{ 1 }{ 2 } \times \left( 2\sqrt{ x }+x+\int{ x } \mathrm{d} x \right)$

Use $\int{ x } \mathrm{d} x=\frac{ {x}^{2} }{ 2 }$ to evaluate the integral

$\frac{ 1 }{ 2 } \times \left( 2\sqrt{ x }+x+\frac{ {x}^{2} }{ 2 } \right)$

Distribute $\frac{ 1 }{ 2 }$ through the parentheses

$\sqrt{ x }+\frac{ 1 }{ 2 }x+\frac{ {x}^{2} }{ 4 }$

Add the constant of integration $C \in ℝ$

$\begin{array} { l }\sqrt{ x }+\frac{ 1 }{ 2 }x+\frac{ {x}^{2} }{ 4 }+C,& C \in ℝ\end{array}$