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

$$\int_{ 0 }^{ 1 } { x }^{ 2 } - \frac{ 1 }{ 2 } \sqrt{ x } d x$$

Evaluate

$$0$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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

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

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. Multiply -\frac{1}{2} times \frac{2x^{\frac{3}{2}}}{3}.

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

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.

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

Simplify.

$$0$$