Solve ∫ (from -2 to 2) of x^3-5 wrt x

$$\int_{ -2 }^{ 2 } { x }^{ 3 } -5 d x$$

Evaluate

$$-20$$

Solution Steps

Evaluate the indefinite integral first.

$$\int x^{3}-5\mathrm{d}x$$

Integrate the sum term by term.

$$\int x^{3}\mathrm{d}x+\int -5\mathrm{d}x$$

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}.

$$\frac{x^{4}}{4}+\int -5\mathrm{d}x$$

Find the integral of -5 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{x^{4}}{4}-5x$$

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.

$$\frac{2^{4}}{4}-5\times 2-\left(\frac{\left(-2\right)^{4}}{4}-5\left(-2\right)\right)$$

Simplify.

$$-20$$