Solve ∫ (from 0 to 30) of 50(30-x) wrt x

$$\int_{ 0 }^{ 30 } 50(30-x) d x$$

Evaluate

$$22500$$

Solution Steps

Use the distributive property to multiply 50 by 30-x.

$$\int _{0}^{30}1500-50x\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int 1500-50x\mathrm{d}x$$

Integrate the sum term by term.

$$\int 1500\mathrm{d}x+\int -50x\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int 1500\mathrm{d}x-50\int x\mathrm{d}x$$

Find the integral of 1500 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$1500x-50\int x\mathrm{d}x$$

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 -50 times \frac{x^{2}}{2}.

$$1500x-25x^{2}$$

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.

$$1500\times 30-25\times 30^{2}-\left(1500\times 0-25\times 0^{2}\right)$$

Simplify.

$$22500$$