Solve ∫ (from -4 to 2) of 12-x-x^2 wrt x

$$\int_{ -4 }^{ 2 } 12-x- { x }^{ 2 } d x$$

Evaluate

$$54$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 12-x-x^{2}\mathrm{d}x$$

Integrate the sum term by term.

$$\int 12\mathrm{d}x+\int -x\mathrm{d}x+\int -x^{2}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int 12\mathrm{d}x-\int x\mathrm{d}x-\int x^{2}\mathrm{d}x$$

Find the integral of 12 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$12x-\int x\mathrm{d}x-\int x^{2}\mathrm{d}x$$

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 -1 times \frac{x^{2}}{2}.

$$12x-\frac{x^{2}}{2}-\int x^{2}\mathrm{d}x$$

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 -1 times \frac{x^{3}}{3}.

$$12x-\frac{x^{2}}{2}-\frac{x^{3}}{3}$$

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.

$$12\times 2-\frac{2^{2}}{2}-\frac{2^{3}}{3}-\left(12\left(-4\right)-\frac{\left(-4\right)^{2}}{2}-\frac{\left(-4\right)^{3}}{3}\right)$$

Simplify.

$$54$$