Solve ∫ (from 5 to 8) of (13-2x) wrt x

$$\int _ { 5 } ^ { 8 } ( 13 - 2 x ) d x$$

Evaluate

$$0$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 13-2x\mathrm{d}x$$

Integrate the sum term by term.

$$\int 13\mathrm{d}x+\int -2x\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int 13\mathrm{d}x-2\int x\mathrm{d}x$$

Find the integral of 13 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$13x-2\int x\mathrm{d}x$$

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

$$13x-x^{2}$$

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.

$$13\times 8-8^{2}-\left(13\times 5-5^{2}\right)$$

Simplify.

$$0$$