Calculate: integral of x^2+cos(x)^3+3 x

Expression: $\int{ {x}^{2}+{\cos\left({x}\right)}^{3}+3 } \mathrm{d} x$

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

$\int{ {x}^{2} } \mathrm{d} x+\int{ {\cos\left({x}\right)}^{3} } \mathrm{d} x+\int{ 3 } \mathrm{d} x$

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

$\frac{ {x}^{3} }{ 3 }+\int{ {\cos\left({x}\right)}^{3} } \mathrm{d} x+\int{ 3 } \mathrm{d} x$

Evaluate the indefinite integral

$\frac{ {x}^{3} }{ 3 }+\sin\left({x}\right)-\frac{ {\sin\left({x}\right)}^{3} }{ 3 }+\int{ 3 } \mathrm{d} x$

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

$\frac{ {x}^{3} }{ 3 }+\sin\left({x}\right)-\frac{ {\sin\left({x}\right)}^{3} }{ 3 }+3x$

Simplify the expression

$\frac{ {x}^{3}-{\sin\left({x}\right)}^{3} }{ 3 }+\sin\left({x}\right)+3x$

Add the constant of integration $C \in ℝ$

$\begin{array} { l }\frac{ {x}^{3}-{\sin\left({x}\right)}^{3} }{ 3 }+\sin\left({x}\right)+3x+C,& C \in ℝ\end{array}$