To evaluate the integral \(\int_{1}^{3}(x^2 - 1)dx\), we will follow a step-by-step approach. ### Step 1: Find the Antiderivative The first step in evaluating a definite integral is to find the antiderivative (also known as the indefinite integral) of the function \(f(x) = x^2 - 1\). The antiderivative of \(x^2\) is \(\frac{x^3}{3}\), and the antiderivative of \(-1\) is \(-x\). Therefore, the antiderivative of \(f(x)\) is: \[ F(x) = \frac{x^3}{3} - x + C \] where \(C\) is the constant of integration, which we do not need for definite integrals. ### Step 2: Evaluate the Definite Integral Now we will evaluate the definite integral from \(1\) to \(3\): \[ \int_{1}^{3}(x^2 - 1)dx = F(3) - F(1) \] #### Calculate \(F(3)\): Substituting \(x = 3\) into \(F(x)\): \[ F(3) = \frac{3^3}{3} - 3 = \frac{27}{3} - 3 = 9 - 3 = 6 \] #### Calculate \(F(1)\): Substituting \(x = 1\) into \(F(x)\): \[ F(1) = \frac{1^3}{3} - 1 = \frac{1}{3} - 1 = \frac{1}{3} - \frac{3}{3} = -\frac{2}{3} \] ### Step 3: Compute the Final Result Now we can find the value of the definite integral: \[ \int_{1}^{3}(x^2 - 1)dx = F(3) - F(1) = 6 - \left(-\frac{2}{3}\right) = 6 + \frac{2}{3} \] To combine these, we convert \(6\) into a fraction with a denominator of \(3\): \[ 6 = \frac{18}{3} \] Thus, \[ 6 + \frac{2}{3} = \frac{18}{3} + \frac{2}{3} = \frac{20}{3} \] ### Conclusion The value of the integral \(\int_{1}^{3}(x^2 - 1)dx\) is \(\frac{20}{3}\). ### Answer Verification Now, let's check the options provided: - A. \(\frac{2}{3}\) - B. \(-\frac{2}{3}\) - C. \(-6\frac{2}{3}\) - D. \(6\frac{2}{3}\) None of the options match \(\frac{20}{3}\). It seems there was a misunderstanding in the options provided. The correct answer is \(\frac{20}{3}\), which is not listed. ### Summary of Key Points - The antiderivative of \(x^2 - 1\) is \(\frac{x^3}{3} - x\). - Evaluating the definite integral involves calculating the antiderivative at the upper and lower limits and subtracting. - The final result of the integral \(\int_{1}^{3}(x^2 - 1)dx\) is \(\frac{20}{3}\). - Always ensure to check the options against your calculated result to confirm correctness. If you have any further questions or need clarification on any step, feel free to ask!