Calculate the integral ∫(2 to 5) (4x - 1) dx. (2023)

Calculate the integral ∫(2 to 5) (4x - 1) dx. (2023)

Step 1: Identify the function to integrate, which is (4x - 1).
Step 2: Find the antiderivative of the function. The antiderivative of (4x - 1) is (2x^2 - x).
Step 3: Set up the definite integral from 2 to 5 using the antiderivative: [2x^2 - x] from 2 to 5.
Step 4: Calculate the value of the antiderivative at the upper limit (5): 2(5^2) - 5 = 2(25) - 5 = 50 - 5 = 45.
Step 5: Calculate the value of the antiderivative at the lower limit (2): 2(2^2) - 2 = 2(4) - 2 = 8 - 2 = 6.
Step 6: Subtract the lower limit result from the upper limit result: 45 - 6 = 39.

Definite Integral – The process of calculating the area under a curve defined by a function over a specific interval.
Fundamental Theorem of Calculus – Relates differentiation and integration, allowing the evaluation of definite integrals using antiderivatives.
Evaluation of Limits – The process of substituting the upper and lower limits into the antiderivative to find the definite integral's value.