Solve ∫ (from 0 to 20) of (x-10)-(50-x) wrt x

$$\int_{ 0 }^{ 20 } (x-10)-(50-x) d x$$

Evaluate

$$-800$$

Solution Steps

To find the opposite of 50-x, find the opposite of each term.

$$\int _{0}^{20}x-10-50-\left(-x\right)\mathrm{d}x$$

The opposite of -x is x.

$$\int _{0}^{20}x-10-50+x\mathrm{d}x$$

Subtract 50 from -10 to get -60.

$$\int _{0}^{20}x-60+x\mathrm{d}x$$

Combine x and x to get 2x.

$$\int _{0}^{20}2x-60\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int 2x-60\mathrm{d}x$$

Integrate the sum term by term.

$$\int 2x\mathrm{d}x+\int -60\mathrm{d}x$$

Factor out the constant in each of the terms.

$$2\int x\mathrm{d}x+\int -60\mathrm{d}x$$

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}.

$$x^{2}+\int -60\mathrm{d}x$$

Find the integral of -60 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$x^{2}-60x$$

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.

$$20^{2}-60\times 20-\left(0^{2}-60\times 0\right)$$

Simplify.

$$-800$$