Solve ∫ (from 2 to 5) of -2x^2-20

$$\int _ { 2 } ^ { 5 } - 2 x ^ { 2 } - 20$$

Evaluate

$$-138$$

Solution Steps

Evaluate the indefinite integral first.

$$\int -2x^{2}-20\mathrm{d}x$$

Integrate the sum term by term.

$$\int -2x^{2}\mathrm{d}x+\int -20\mathrm{d}x$$

Factor out the constant in each of the terms.

$$-2\int x^{2}\mathrm{d}x+\int -20\mathrm{d}x$$

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 -2 times \frac{x^{3}}{3}.

$$-\frac{2x^{3}}{3}+\int -20\mathrm{d}x$$

Find the integral of -20 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$-\frac{2x^{3}}{3}-20x$$

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.

$$-\frac{2}{3}\times 5^{3}-20\times 5-\left(-\frac{2}{3}\times 2^{3}-20\times 2\right)$$

Simplify.

$$-138$$