Solve ∫ (from 12 to 20) of (80-4x) wrt x=

$$\int _ { 12 } ^ { 20 } ( 80 - 4 x ) d x =$$

Evaluate

$$128$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 80-4x\mathrm{d}x$$

Integrate the sum term by term.

$$\int 80\mathrm{d}x+\int -4x\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int 80\mathrm{d}x-4\int x\mathrm{d}x$$

Find the integral of 80 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$80x-4\int x\mathrm{d}x$$

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 -4 times \frac{x^{2}}{2}.

$$80x-2x^{2}$$

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.

$$80\times 20-2\times 20^{2}-\left(80\times 12-2\times 12^{2}\right)$$

Simplify.

$$128$$