Solve ∫ (from 10 to 80) of 1/sqrt[5]{x^4 wrt x

$$\int _ { 10 } ^ { 80 } \frac { 1 } { \sqrt[ 5 ] { x ^ { 4 } } } d x$$

Evaluate

$$5\left(\sqrt[5]{80}-\sqrt[5]{10}\right)\approx 4.086778378$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \frac{1}{\sqrt[5]{x^{4}}}\mathrm{d}x$$

Rewrite \frac{1}{x^{\frac{4}{5}}} as x^{-\frac{4}{5}}. Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{-\frac{4}{5}}\mathrm{d}x with \frac{x^{\frac{1}{5}}}{\frac{1}{5}}. Simplify and convert from exponential to radical form.

$$5\sqrt[5]{x}$$

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.

$$5\times 80^{\frac{1}{5}}-5\times 10^{\frac{1}{5}}$$

Simplify.

$$5\sqrt[5]{80}-5\sqrt[5]{10}$$