Question

In Exercises \(27-40\) , evaluate each integral using Part 2 of the Fundamental Theorem. Support your answer with NINT if you are unsure. $$\int_{0}^{\pi / 3} 4 \sec x \tan x d x$$

Short answer

The value of the integral is 4.

Step-by-step solution

Identify the Antiderivative

Identify the antiderivative of \(4 \sec x \tan x\). The integral of \(\sec x \tan x\) is \(\sec x\). Therefore, the antiderivative of \(4 \sec x \tan x\) is \(4 \sec x\).

Apply the Fundamental Theorem of Calculus

Now, apply the Fundamental Theorem of Calculus, which states that the definite integral of a function is equal to its antiderivative evaluated at the upper limit of integration minus its antiderivative evaluated at the lower limit. Applying this to the current problem yields: \(4 \sec(\pi / 3 ) - 4 \sec (0)\).

Evaluate the Expression

Substitute the values into the secant function: \sec(\pi / 3) is equal to 2 and \sec(0) is equal to 1. So, you get \(4 * 2 - 4*1 = 4\).