Introduction:
The Trapezoidal Rule is a numerical integration method that approximates the definite integral of a function by dividing the area under the curve into trapezoids. This method is simple to implement and provides a good approximation for smooth functions.

Explanation:
The Trapezoidal Rule can give an exact value of the integral when the integrand is a linear function. In this case, the function can be represented by a straight line, and the area under the curve can be accurately calculated using the trapezoidal approximation.

Reasoning:
The Trapezoidal Rule approximates the area under the curve by dividing it into trapezoids. Each trapezoid is formed by connecting two adjacent points on the curve with a straight line segment. The area of each trapezoid is then calculated by taking the average of the heights of the two adjacent points and multiplying it by the width of the trapezoid.

When the integrand is a linear function, the curve is a straight line. In this case, the Trapezoidal Rule accurately approximates the area under the curve by dividing it into trapezoids with equal widths. Since the function is linear, the heights of the trapezoids remain constant throughout the interval, and the trapezoidal approximation becomes exact.

Example:
Let's consider the linear function f(x) = 2x + 3 over the interval [1, 5]. Using the Trapezoidal Rule, we divide the interval into n subintervals and approximate the integral as follows:

∫[1, 5] (2x + 3) dx ≈ Δx/2 * [f(x0) + 2f(x1) + 2f(x2) + ... + 2f(xn-1) + f(xn)]

where Δx = (b - a)/n is the width of each subinterval, x0 = 1, xn = 5, and xi = a + iΔx.

Since the integrand is a linear function, the function values f(xi) = 2xi + 3 remain constant throughout the interval. Therefore, the Trapezoidal Rule becomes exact, and the approximation becomes the exact value of the integral.

Conclusion:
The Trapezoidal Rule gives an exact value of the integral when the integrand is a linear function. This is because the Trapezoidal Rule accurately approximates the area under the curve by dividing it into trapezoids with equal widths. For linear functions, the heights of the trapezoids remain constant, resulting in an exact approximation of the integral.