Solve ∫ (from - 2 to 2) of (1-x) wrt x

$$\int _ { - 2 } ^ { 2 } ( 1 - x ) d x$$

Evaluate

$$4$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 1-x\mathrm{d}x$$

Integrate the sum term by term.

$$\int 1\mathrm{d}x+\int -x\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int 1\mathrm{d}x-\int x\mathrm{d}x$$

Find the integral of 1 using the table of common integrals rule \int a\mathrm{d}x=ax.

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

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

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

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.

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

Simplify.

$$4$$