Solve ∫ (from 1 to 3) of 1/x^5 wrt x

$$\int _ { 1 } ^ { 3 } \frac { 1 } { x ^ { 5 } } d x$$

Evaluate

$$\frac{20}{81}\approx 0.24691358$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \frac{1}{x^{5}}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int \frac{1}{x^{5}}\mathrm{d}x with -\frac{1}{4x^{4}}.

$$-\frac{1}{4x^{4}}$$

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.

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

Simplify.

$$\frac{20}{81}$$