How do you evaluate the definite integral ∫(2x-1)dx from [1,3]?

To evaluate the definite integral ∫(2x-1)dx from 1 to 3, follow these steps:

Step 1: Find the Antiderivative

The first step is to determine the antiderivative of the function 2x - 1. The antiderivative is found by integrating:

• ∫(2x)dx = x²
• ∫(-1)dx = -x

Combining these results, the antiderivative of 2x - 1 is:

Step 2: Evaluate the Antiderivative at the Bounds

Next, we evaluate this antiderivative at the upper and lower limits of the integral:

• F(3) = 3² - 3 = 9 - 3 = 6
• F(1) = 1² - 1 = 1 - 1 = 0

Step 3: Calculate the Definite Integral

Now, subtract the value of the antiderivative at the lower limit from the value at the upper limit:

Final Result

The value of the definite integral ∫(2x-1)dx from 1 to 3 is 6.