Solve ∫ (from 1 to 3) of |t^2-t-2|

$$\int _ { 1 } ^ { 3 } | t ^ { 2 } - t - 2 |$$

Evaluate

$$2|t-2||t+1|$$

Solution Steps

Evaluate the indefinite integral first.

$$\int |t^{2}-t-2|\mathrm{d}x$$

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

$$|t^{2}-t-2|x$$

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.

$$|t^{2}-t-2|\times 3-|t^{2}-t-2|$$

Simplify.

$$2|-2+t||1+t|$$

Differentiate w.r.t. t

$$2\left(|t-2|sign(t+1)+sign(t-2)|t+1|\right)$$