Solve ∫ (from - 5 to 5) of (t^2-1)dt

$$\int _ { - 5 } ^ { 5 } ( t ^ { 2 } - 1 ) d t$$

Evaluate

$$\frac{220}{3}\approx 73.333333333$$

Solution Steps

Evaluate the indefinite integral first.

$$\int t^{2}-1\mathrm{d}t$$

Integrate the sum term by term.

$$\int t^{2}\mathrm{d}t+\int -1\mathrm{d}t$$

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

$$\frac{t^{3}}{3}+\int -1\mathrm{d}t$$

Find the integral of -1 using the table of common integrals rule \int a\mathrm{d}t=at.

$$\frac{t^{3}}{3}-t$$

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.

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

Simplify.

$$\frac{220}{3}$$