Solve ∫ (from 0 to 30) of (3900-120/1+5x) wrt x

$$\int _ { 0 } ^ { 30 } ( 3900 - \frac { 120 } { 1 + 5 } x ) d x$$

Evaluate

$$108000$$

Solution Steps

Add 1 and 5 to get 6.

$$\int _{0}^{30}3900-\frac{120}{6}x\mathrm{d}x$$

Divide 120 by 6 to get 20.

$$\int _{0}^{30}3900-20x\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int 3900-20x\mathrm{d}x$$

Integrate the sum term by term.

$$\int 3900\mathrm{d}x+\int -20x\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int 3900\mathrm{d}x-20\int x\mathrm{d}x$$

Find the integral of 3900 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$3900x-20\int x\mathrm{d}x$$

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 -20 times \frac{x^{2}}{2}.

$$3900x-10x^{2}$$

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.

$$3900\times 30-10\times 30^{2}-\left(3900\times 0-10\times 0^{2}\right)$$

Simplify.

$$108000$$