Solve ∫ (from 0 to 2) of x^6-x^3 wrt x

$$\int _ { 0 } ^ { 2 } x ^ { 6 } - x ^ { 3 } d x$$

Evaluate

$$\frac{100}{7}\approx 14.285714286$$

Solution Steps

Evaluate the indefinite integral first.

$$\int x^{6}-x^{3}\mathrm{d}x$$

Integrate the sum term by term.

$$\int x^{6}\mathrm{d}x+\int -x^{3}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int x^{6}\mathrm{d}x-\int x^{3}\mathrm{d}x$$

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}.

$$\frac{x^{7}}{7}-\int x^{3}\mathrm{d}x$$

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

$$\frac{x^{7}}{7}-\frac{x^{4}}{4}$$

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.

$$\frac{2^{7}}{7}-\frac{2^{4}}{4}-\left(\frac{0^{7}}{7}-\frac{0^{4}}{4}\right)$$

Simplify.

$$\frac{100}{7}$$