Solve ∫ (from 0 to 4) of (-18-18x-4.5x^2)(-x) wrt x

$$\int_{ 0 }^{ 4 } (-18-18x-4.5 { x }^{ 2 } )(-x) d x$$

Evaluate

$$816$$

Solution Steps

Use the distributive property to multiply -18-18x-4.5x^{2} by -x.

$$\int _{0}^{4}-18\left(-x\right)-18x\left(-x\right)-4.5x^{2}\left(-x\right)\mathrm{d}x$$

Multiply -18 and -1 to get 18.

$$\int _{0}^{4}18x-18x\left(-x\right)-4.5x^{2}\left(-x\right)\mathrm{d}x$$

Multiply -18 and -1 to get 18.

$$\int _{0}^{4}18x+18xx-4.5x^{2}\left(-x\right)\mathrm{d}x$$

Multiply x and x to get x^{2}.

$$\int _{0}^{4}18x+18x^{2}-4.5x^{2}\left(-x\right)\mathrm{d}x$$

Multiply -4.5 and -1 to get 4.5.

$$\int _{0}^{4}18x+18x^{2}+4.5x^{2}x\mathrm{d}x$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$\int _{0}^{4}18x+18x^{2}+4.5x^{3}\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int 18x+18x^{2}+\frac{9x^{3}}{2}\mathrm{d}x$$

Integrate the sum term by term.

$$\int 18x\mathrm{d}x+\int 18x^{2}\mathrm{d}x+\int \frac{9x^{3}}{2}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$18\int x\mathrm{d}x+18\int x^{2}\mathrm{d}x+\frac{9\int x^{3}\mathrm{d}x}{2}$$

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 18 times \frac{x^{2}}{2}.

$$9x^{2}+18\int x^{2}\mathrm{d}x+\frac{9\int x^{3}\mathrm{d}x}{2}$$

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

$$9x^{2}+6x^{3}+\frac{9\int x^{3}\mathrm{d}x}{2}$$

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 4.5 times \frac{x^{4}}{4}.

$$9x^{2}+6x^{3}+\frac{9x^{4}}{8}$$

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.

$$9\times 4^{2}+6\times 4^{3}+\frac{9}{8}\times 4^{4}-\left(9\times 0^{2}+6\times 0^{3}+\frac{9}{8}\times 0^{4}\right)$$

Simplify.

$$816$$