Solve ∫ _{1/2}^1(1/u^4-1/u^2)du

$$\int _ { \frac { 1 } { 2 } } ^ { 1 } ( \frac { 1 } { u ^ { 4 } } - \frac { 1 } { u ^ { 2 } } ) d u$$

Evaluate

$$\frac{4}{3}\approx 1.333333333$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \frac{1}{u^{4}}-\frac{1}{u^{2}}\mathrm{d}u$$

Integrate the sum term by term.

$$\int \frac{1}{u^{4}}\mathrm{d}u+\int -\frac{1}{u^{2}}\mathrm{d}u$$

Factor out the constant in each of the terms.

$$\int \frac{1}{u^{4}}\mathrm{d}u-\int \frac{1}{u^{2}}\mathrm{d}u$$

Since \int u^{k}\mathrm{d}u=\frac{u^{k+1}}{k+1} for k\neq -1, replace \int \frac{1}{u^{4}}\mathrm{d}u with -\frac{1}{3u^{3}}.

$$-\frac{1}{3u^{3}}-\int \frac{1}{u^{2}}\mathrm{d}u$$

Since \int u^{k}\mathrm{d}u=\frac{u^{k+1}}{k+1} for k\neq -1, replace \int \frac{1}{u^{2}}\mathrm{d}u with -\frac{1}{u}. Multiply -1 times -\frac{1}{u}.

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

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.

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

Simplify.

$$\frac{4}{3}$$