Solve ∫ (x^2+3x+3sqrt{x}) wrt x

$$\int{ ( { x }^{ 2 } +3x+3 \sqrt{ x } ) }d x$$

Evaluate

$$\frac{x^{3}}{3}+\frac{3x^{2}}{2}+2x^{\frac{3}{2}}+С$$

Solution Steps

Integrate the sum term by term.

$$\int x^{2}\mathrm{d}x+\int 3x\mathrm{d}x+\int 3\sqrt{x}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int x^{2}\mathrm{d}x+3\int x\mathrm{d}x+3\int \sqrt{x}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{2}\mathrm{d}x with \frac{x^{3}}{3}.

$$\frac{x^{3}}{3}+3\int x\mathrm{d}x+3\int \sqrt{x}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x\mathrm{d}x with \frac{x^{2}}{2}. Multiply 3 times \frac{x^{2}}{2}.

$$\frac{x^{3}}{3}+\frac{3x^{2}}{2}+3\int \sqrt{x}\mathrm{d}x$$

Rewrite \sqrt{x} as x^{\frac{1}{2}}. Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{\frac{1}{2}}\mathrm{d}x with \frac{x^{\frac{3}{2}}}{\frac{3}{2}}. Simplify. Multiply 3 times \frac{2x^{\frac{3}{2}}}{3}.

$$\frac{x^{3}}{3}+\frac{3x^{2}}{2}+2x^{\frac{3}{2}}$$

If F\left(x\right) is an antiderivative of f\left(x\right), then the set of all antiderivatives of f\left(x\right) is given by F\left(x\right)+C. Therefore, add the constant of integration C\in \mathrm{R} to the result.

$$\frac{x^{3}}{3}+\frac{3x^{2}}{2}+2x^{\frac{3}{2}}+С$$

Differentiate w.r.t. x

$$x^{2}+3x+3\sqrt{x}$$