Solve ∫ (sqrt{x}+x/2) wrt x

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

Evaluate

$$\frac{x^{2}}{4}+\frac{2x^{\frac{3}{2}}}{3}+С$$

Solution Steps

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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.

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

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 \frac{1}{2} times \frac{x^{2}}{2}.

$$\frac{2x^{\frac{3}{2}}}{3}+\frac{x^{2}}{4}$$

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.

$$\frac{2x^{\frac{3}{2}}}{3}+\frac{x^{2}}{4}+С$$

Differentiate w.r.t. x

$$\frac{x}{2}+\sqrt{x}$$