Solve ∫ (from - 1 to 8) of sqrt[3]{x wrt x

$$\int _ { - 1 } ^ { 8 } \sqrt[ 3 ] { x } d x$$

Evaluate

$$\frac{45}{4}=11.25$$

Solution Steps

Evaluate the indefinite integral first.

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

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.

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

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.

$$\frac{3}{4}\times 8^{\frac{4}{3}}-\frac{3}{4}\left(-1\right)^{\frac{4}{3}}$$

Simplify.

$$\frac{45}{4}$$