Solve ∫ (from 2 to 4) of (-1/2x+2) wrt x

$$\int _ { 2 } ^ { 4 } ( - \frac { 1 } { 2 } x + 2 ) d x$$

Evaluate

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Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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}.

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

Find the integral of 2 using the table of common integrals rule \int a\mathrm{d}x=ax.

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

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.

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

Simplify.

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