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

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

Evaluate

$$0$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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}.

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

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

$$-\frac{x^{4}}{4}-\frac{x^{2}}{2}$$

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.

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

Simplify.

$$0$$