Solve ∫ (from 0 to 2) of (-1/2x+1)

$$\int _ { 0 } ^ { 2 } ( - \frac { 1 } { 2 } x + 1 )$$

Evaluate

$$1$$

Solution Steps

Evaluate the indefinite integral first.

$$\int -\frac{x}{2}+1\mathrm{d}x$$

Integrate the sum term by term.

$$\int -\frac{x}{2}\mathrm{d}x+\int 1\mathrm{d}x$$

Factor out the constant in each of the terms.

$$-\frac{\int x\mathrm{d}x}{2}+\int 1\mathrm{d}x$$

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 -\frac{1}{2} times \frac{x^{2}}{2}.

$$-\frac{x^{2}}{4}+\int 1\mathrm{d}x$$

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

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

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.

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

Simplify.

$$1$$