Solve ∫ (from 0 to 2) of (x-sqrt[3]{x}) wrt x

$$\int _ { 0 } ^ { 2 } ( x - \sqrt[ 3 ] { x } ) d x$$

Evaluate

$$-\frac{3\sqrt[3]{2}}{2}+2\approx 0.110118425$$

Solution Steps

Evaluate the indefinite integral first.

$$\int x-\sqrt[3]{x}\mathrm{d}x$$

Integrate the sum term by term.

$$\int x\mathrm{d}x+\int -\sqrt[3]{x}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int x\mathrm{d}x-\int \sqrt[3]{x}\mathrm{d}x$$

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}.

$$\frac{x^{2}}{2}-\int \sqrt[3]{x}\mathrm{d}x$$

Rewrite \sqrt[3]{x} as x^{\frac{1}{3}}. Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{\frac{1}{3}}\mathrm{d}x with \frac{x^{\frac{4}{3}}}{\frac{4}{3}}. Simplify. Multiply -1 times \frac{3x^{\frac{4}{3}}}{4}.

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

The definite integral is the antiderivative of the expression evaluated at the upper limit of integration minus the antiderivative evaluated at the lower limit of integration.

$$\frac{2^{2}}{2}-\frac{3}{4}\times 2^{\frac{4}{3}}-\left(\frac{0^{2}}{2}-\frac{3}{4}\times 0^{\frac{4}{3}}\right)$$

Simplify.

$$2-\frac{3\sqrt[3]{2}}{2}$$