Solve ∫ (from 1 to 2) of frac{4sqrt[3]{x}}{3 wrt x

$$\int _ { 1 } ^ { 2 } \frac { 4 \sqrt[ 3 ] { x } } { 3 } d x$$

Evaluate

$$2^{\frac{4}{3}}-1\approx 1.5198421$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \frac{4\sqrt[3]{x}}{3}\mathrm{d}x$$

Factor out the constant using \int af\left(x\right)\mathrm{d}x=a\int f\left(x\right)\mathrm{d}x.

$$\frac{4}{3}\int \sqrt[3]{x}\mathrm{d}x$$

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.

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

Simplify.

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

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.

$$2^{\frac{4}{3}}-1^{\frac{4}{3}}$$

Simplify.

$$2\sqrt[3]{2}-1$$