Solve ∫ (from - 1 to 3) of (x^3-1) wrt x

$$\int _ { - 1 } ^ { 3 } ( x ^ { 3 } - 1 ) d x$$

Evaluate

$$16$$

Solution Steps

Evaluate the indefinite integral first.

$$\int x^{3}-1\mathrm{d}x$$

Integrate the sum term by term.

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

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}.

$$\frac{x^{4}}{4}+\int -1\mathrm{d}x$$

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

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

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.

$$\frac{3^{4}}{4}-3-\left(\frac{\left(-1\right)^{4}}{4}-\left(-1\right)\right)$$

Simplify.

$$16$$