Solve ∫ (from -2 to 3) of x((x^2)-(x+6)) wrt x

$$\int_{ -2 }^{ 3 } x(( { x }^{ 2 } )-(x+6)) d x$$

Evaluate

$$-\frac{125}{12}\approx -10.416666667$$

Solution Steps

To find the opposite of x+6, find the opposite of each term.

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

Use the distributive property to multiply x by x^{2}-x-6.

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

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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

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

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

$$\frac{x^{4}}{4}-\frac{x^{3}}{3}-6\int x\mathrm{d}x$$

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 -6 times \frac{x^{2}}{2}.

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

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.

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

Simplify.

$$-\frac{125}{12}$$