Solve ∫ (from 0 to 3) of [(x+2)-x(x+2)(x+4)] wrt x

$$\int _ { 0 } ^ { 3 } [ ( x + 2 ) - x ( x + 2 ) ( x + 4 ) ] d x$$

Evaluate

$$-\frac{399}{4}=-99.75$$

Solution Steps

Use the distributive property to multiply x by x+2.

$$\int _{0}^{3}x+2-\left(x^{2}+2x\right)\left(x+4\right)\mathrm{d}x$$

Apply the distributive property by multiplying each term of x^{2}+2x by each term of x+4.

$$\int _{0}^{3}x+2-\left(x^{3}+4x^{2}+2x^{2}+8x\right)\mathrm{d}x$$

Combine 4x^{2} and 2x^{2} to get 6x^{2}.

$$\int _{0}^{3}x+2-\left(x^{3}+6x^{2}+8x\right)\mathrm{d}x$$

To find the opposite of x^{3}+6x^{2}+8x, find the opposite of each term.

$$\int _{0}^{3}x+2-x^{3}-6x^{2}-8x\mathrm{d}x$$

Combine x and -8x to get -7x.

$$\int _{0}^{3}-7x+2-x^{3}-6x^{2}\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int -7x+2-x^{3}-6x^{2}\mathrm{d}x$$

Integrate the sum term by term.

$$\int -7x\mathrm{d}x+\int 2\mathrm{d}x+\int -x^{3}\mathrm{d}x+\int -6x^{2}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$-7\int x\mathrm{d}x+\int 2\mathrm{d}x-\int x^{3}\mathrm{d}x-6\int x^{2}\mathrm{d}x$$

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 -7 times \frac{x^{2}}{2}.

$$-\frac{7x^{2}}{2}+\int 2\mathrm{d}x-\int x^{3}\mathrm{d}x-6\int x^{2}\mathrm{d}x$$

Find the integral of 2 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$-\frac{7x^{2}}{2}+2x-\int x^{3}\mathrm{d}x-6\int x^{2}\mathrm{d}x$$

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 -1 times \frac{x^{4}}{4}.

$$-\frac{7x^{2}}{2}+2x-\frac{x^{4}}{4}-6\int x^{2}\mathrm{d}x$$

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 -6 times \frac{x^{3}}{3}.

$$-\frac{7x^{2}}{2}+2x-\frac{x^{4}}{4}-2x^{3}$$

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.

$$-\frac{7}{2}\times 3^{2}+2\times 3-\frac{3^{4}}{4}-2\times 3^{3}-\left(-\frac{7}{2}\times 0^{2}+2\times 0-\frac{0^{4}}{4}-2\times 0^{3}\right)$$

Simplify.

$$-\frac{399}{4}$$