Solve ∫ (from 0 to 1) of (2sqrt{x}-x) wrt x

$$\int _ { 0 } ^ { 1 } ( 2 \sqrt { x } - x ) d x$$

Evaluate

$$\frac{5}{6}\approx 0.833333333$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 2\sqrt{x}-x\mathrm{d}x$$

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

$$2\int \sqrt{x}\mathrm{d}x-\int x\mathrm{d}x$$

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 2 times \frac{2x^{\frac{3}{2}}}{3}.

$$\frac{4x^{\frac{3}{2}}}{3}-\int x\mathrm{d}x$$

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

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

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.

$$\frac{4}{3}\times 1^{\frac{3}{2}}-\frac{1^{2}}{2}-\left(\frac{4}{3}\times 0^{\frac{3}{2}}-\frac{0^{2}}{2}\right)$$

Simplify.

$$\frac{5}{6}$$