Solve ∫ (from 0 to 400) of (88-0.1x-0.0003x^2) wrt x

$$\int _ { 0 } ^ { 400 } ( 88 - 0.1 x - 0.0003 x ^ { 2 } ) d x$$

Evaluate

$$20800$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 88-\frac{x}{10}-\frac{3x^{2}}{10000}\mathrm{d}x$$

Integrate the sum term by term.

$$\int 88\mathrm{d}x+\int -\frac{x}{10}\mathrm{d}x+\int -\frac{3x^{2}}{10000}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int 88\mathrm{d}x-\frac{\int x\mathrm{d}x}{10}-\frac{3\int x^{2}\mathrm{d}x}{10000}$$

Find the integral of 88 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$88x-\frac{\int x\mathrm{d}x}{10}-\frac{3\int x^{2}\mathrm{d}x}{10000}$$

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 -0.1 times \frac{x^{2}}{2}.

$$88x-\frac{x^{2}}{20}-\frac{3\int x^{2}\mathrm{d}x}{10000}$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{2}\mathrm{d}x with \frac{x^{3}}{3}. Multiply -0.0003 times \frac{x^{3}}{3}.

$$88x-\frac{x^{2}}{20}-\frac{x^{3}}{10000}$$

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.

$$88\times 400-\frac{400^{2}}{20}-\frac{400^{3}}{10000}-\left(88\times 0-\frac{0^{2}}{20}-\frac{0^{3}}{10000}\right)$$

Simplify.

$$20800$$