Solve ∫ (from 0 to frac { 1) of {2}}x^6-x^4 wrt x

$$\int _ { 0 } ^ { \frac { 1 } { 2 } } x ^ { 6 } - x ^ { 4 } d x$$

Evaluate

$$-\frac{23}{4480}\approx -0.005133929$$

Solution Steps

Evaluate the indefinite integral first.

$$\int x^{6}-x^{4}\mathrm{d}x$$

Integrate the sum term by term.

$$\int x^{6}\mathrm{d}x+\int -x^{4}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int x^{6}\mathrm{d}x-\int x^{4}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{6}\mathrm{d}x with \frac{x^{7}}{7}.

$$\frac{x^{7}}{7}-\int x^{4}\mathrm{d}x$$

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

$$\frac{x^{7}}{7}-\frac{x^{5}}{5}$$

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.

$$\frac{\left(\frac{1}{2}\right)^{7}}{7}-\frac{\left(\frac{1}{2}\right)^{5}}{5}-\left(\frac{0^{7}}{7}-\frac{0^{5}}{5}\right)$$

Simplify.

$$-\frac{23}{4480}$$