Question

In Exercises \(27-40\) , evaluate each integral using Part 2 of the Fundamental Theorem. Support your answer with NINT if you are unsure. $$\int_{1 / 2}^{3}\left(2-\frac{1}{x}\right) d x$$

Short answer

The value of the integral is \((6 - ln(3)) - (1 - ln( \frac{1}{2}))\).

Step-by-step solution

Identify the Function

The function given in the integral that needs to be evaluated is \(2 - \frac{1}{x}\). The limits of integration are from \( \frac{1}{2} \) to \( 3 \).

Find the Anti-Derivative

The anti-derivative of \(2 - \frac{1}{x}\) is found by integrating term by term. The integral of \(2\) with respect to \(x\) is \(2x\), and the integral of \(- \frac{1}{x}\) is \(-ln|x|\). Therefore, the anti-derivative of \(2 - \frac{1}{x}\) is \(2x - ln|x|\).

Apply the Fundamental Theorem of Calculus Part 2

According to the Fundamental Theorem of Calculus Part 2, the definite integral of a function from \(a\) to \(b\) is equal to its anti-derivative evaluated at \(b\) minus its anti-derivative evaluated at \(a\). Here, \(a = \frac{1}{2}\) and \(b = 3\). So, plug in these values into \(2x - ln|x|\) to get the answer.

Evaluate at the Upper Limit

The anti-derivative \(2x - ln|x|\) evaluated at \(b = 3\) is \(2*3 - ln|3|\), which equals \(6 - ln(3)\).

Evaluate at the Lower Limit

The anti-derivative \(2x - ln|x|\) evaluated at \(a = \frac{1}{2}\) is \(2* \frac{1}{2} - ln| \frac{1}{2}|\), which equals \(1 - ln( \frac{1}{2})\).

Subtract the Results

Lastly, subtract the result from Step 5 from the result from Step 4 to find the value of the integral. So, we have \((6 - ln(3)) - (1 - ln( \frac{1}{2}))\).