Solve ∫ (from 0 to 1) of (sqrt{x}+sqrt[3]{x^2}) wrt x=

$$\int _ { 0 } ^ { 1 } ( \sqrt { x } + \sqrt[ 3 ] { x ^ { 2 } } ) d x =$$

Evaluate

$$\frac{19}{15}\approx 1.266666667$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

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.

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

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{\frac{2}{3}}\mathrm{d}x with \frac{3x^{\frac{5}{3}}}{5}.

$$\frac{2x^{\frac{3}{2}}}{3}+\frac{3x^{\frac{5}{3}}}{5}$$

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.

$$\frac{2}{3}\times 1^{\frac{3}{2}}+\frac{3}{5}\times 1^{\frac{5}{3}}-\left(\frac{2}{3}\times 0^{\frac{3}{2}}+\frac{3}{5}\times 0^{\frac{5}{3}}\right)$$

Simplify.

$$\frac{19}{15}$$