Questions: Evaluate the integral. HINT [See Example 2.] [ int-1^1left(x^2+5right) d x ]

Solution Steps

To evaluate the definite integral of the function \( f(x) = x^2 + 5 \) from \(-1\) to \(1\), we will find the antiderivative of the function and then apply the Fundamental Theorem of Calculus. This involves calculating the antiderivative, evaluating it at the upper and lower limits, and finding the difference.

We need to evaluate the definite integral: \[ \int_{-1}^{1} \left( x^2 + 5 \right) \, dx \]

The antiderivative of the function \( f(x) = x^2 + 5 \) is: \[ F(x) = \frac{x^3}{3} + 5x \]

Now we evaluate \( F(x) \) at the upper limit \( x = 1 \) and the lower limit \( x = -1 \): \[ F(1) = \frac{1^3}{3} + 5(1) = \frac{1}{3} + 5 = \frac{1}{3} + \frac{15}{3} = \frac{16}{3} \] \[ F(-1) = \frac{(-1)^3}{3} + 5(-1) = -\frac{1}{3} - 5 = -\frac{1}{3} - \frac{15}{3} = -\frac{16}{3} \]

Now, we find the value of the definite integral by subtracting the lower limit evaluation from the upper limit evaluation: \[ \int_{-1}^{1} \left( x^2 + 5 \right) \, dx = F(1) - F(-1) = \frac{16}{3} - \left(-\frac{16}{3}\right) = \frac{16}{3} + \frac{16}{3} = \frac{32}{3} \]

Final Answer