Solve ∫ (from 0 to 4) of (2x^2-5.25x)(1-0.125(x)) wrt x

$$\int_{ 0 }^{ 4 } (2 { x }^{ 2 } -5.25x)(1-0.125(x)) d x$$

Evaluate

$$-\frac{4}{3}\approx -1.333333333$$

Solution Steps

Use the distributive property to multiply 2x^{2}-5.25x by 1-0.125x and combine like terms.

$$\int _{0}^{4}2.65625x^{2}-0.25x^{3}-5.25x\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int \frac{85x^{2}}{32}-\frac{x^{3}}{4}-\frac{21x}{4}\mathrm{d}x$$

Integrate the sum term by term.

$$\int \frac{85x^{2}}{32}\mathrm{d}x+\int -\frac{x^{3}}{4}\mathrm{d}x+\int -\frac{21x}{4}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\frac{85\int x^{2}\mathrm{d}x}{32}-\frac{\int x^{3}\mathrm{d}x}{4}-\frac{21\int x\mathrm{d}x}{4}$$

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 2.65625 times \frac{x^{3}}{3}.

$$\frac{85x^{3}}{96}-\frac{\int x^{3}\mathrm{d}x}{4}-\frac{21\int x\mathrm{d}x}{4}$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{3}\mathrm{d}x with \frac{x^{4}}{4}. Multiply -0.25 times \frac{x^{4}}{4}.

$$\frac{85x^{3}}{96}-\frac{x^{4}}{16}-\frac{21\int x\mathrm{d}x}{4}$$

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 -5.25 times \frac{x^{2}}{2}.

$$\frac{85x^{3}}{96}-\frac{x^{4}}{16}-\frac{21x^{2}}{8}$$

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.

$$\frac{85}{96}\times 4^{3}-\frac{4^{4}}{16}-\frac{21}{8}\times 4^{2}-\left(\frac{85}{96}\times 0^{3}-\frac{0^{4}}{16}-\frac{21}{8}\times 0^{2}\right)$$

Simplify.

$$-\frac{4}{3}$$