Step by step

1. • Integrate the sum term by term
   • indefinite integral of int 2x raised to 3 dx+ indefinite integral of int xdx$\int 2x^{3}\mathrm{d}x+\int x\mathrm{d}x$
2. • Factor out the constant in each of the terms
   • 2 indefinite integral of int x raised to 3 dx+ indefinite integral of int xdx$2\int x^{3}\mathrm{d}x+\int x\mathrm{d}x$
3. • Since indefinite integral of int x raised to k dx= fraction start x raised to (k+1) } over k+1 fraction end }$\int x^{k}\mathrm{d}x=\frac{x^{\left(k+1\right)}}{k+1}$ for k is not equal to minus 1k ≠ -1, replace indefinite integral of int x raised to 3 dx$\int x^{3}\mathrm{d}x$ with fraction start x raised to 4 } over 4 fraction end }$\frac{x^{4}}{4}$.
   • fraction start x raised to 4 } over 2 fraction end + indefinite integral of int xdx$\dfrac{x^{4}}{2}+\int x\mathrm{d}x$
4. • Since indefinite integral of int x raised to k dx= fraction start x raised to (k+1) } over k+1 fraction end }$\int x^{k}\mathrm{d}x=\frac{x^{\left(k+1\right)}}{k+1}$ for k is not equal to minus 1k ≠ -1, replace indefinite integral of int xdx$\int x\mathrm{d}x$ with fraction start x raised to 2 } over 2 fraction end }$\frac{x^{2}}{2}$.
   • fraction start x raised to 4 +x raised to 2 } over 2 fraction end }$\dfrac{x^{4}+x^{2}}{2}$
5. • If F(x) is an antiderivative of f(x), then the set of all antiderivatives of f(x) is given by F(x)+C. Therefore, add the real constant of integration C to the result.
   • fraction start x raised to 4 } over 2 fraction end + fraction start x raised to 2 } over 2 fraction end +С$\dfrac{x^{4}}{2}+\dfrac{x^{2}}{2}+С$