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1. \int_{-1}^{1} x^{3} - x\, dx

   First find an antiderivative F, then evaluate F(b) − F(a).

2. \int x^{3} - x\, dx = \int x^{3}\, dx + \int - x\, dx

   The integral of a sum is the sum of the integrals.

3. \int x^{3}\, dx = \frac{x^{4}}{4}

   Power rule: ∫xⁿ dx = xⁿ⁺¹/(n+1) (n ≠ −1).

4. \int - x\, dx = -1 \int x\, dx

   Pull the constant -1 out of the integral.

5. \int x\, dx = \frac{x^{2}}{2}

   Power rule: ∫xⁿ dx = xⁿ⁺¹/(n+1) (n ≠ −1).

6. F(1) - F(-1) = \left(- \frac{1}{4}\right) - \left(- \frac{1}{4}\right)

   Fundamental theorem of calculus: plug in the limits.

7. = 0

   Simplify.