Calculate The Integral 1x2 11x 28dx

How to Calculate the Integral

Calculating the integral of 1/x² + 11x - 28 involves breaking the function into its component parts and integrating each term separately. Here's the step-by-step process:

1. Identify the antiderivative for each term in the integrand.
2. Combine the antiderivatives to form the general antiderivative.
3. Apply the limits of integration to find the definite integral.

The antiderivative of 1/x² is -1/x, the antiderivative of 11x is 11x²/2, and the antiderivative of -28 is -28x. Combining these gives the general antiderivative: -1/x + 11x²/2 - 28x + C, where C is the constant of integration.

The Formula

This formula represents the area under the curve of the function between the specified limits. The result will depend on the values of a and b you choose.

Worked Example

Let's calculate the integral from x=1 to x=2:

1. Evaluate the antiderivative at x=2: -1/2 + 11(4)/2 - 56 = -0.5 + 22 - 56 = -34.5
2. Evaluate the antiderivative at x=1: -1 + 11(1)/2 - 28 = -1 + 5.5 - 28 = -23.5
3. Subtract the lower limit from the upper limit: -34.5 - (-23.5) = -11

The integral from 1 to 2 of (1/x² + 11x - 28) dx is -11.

Interpreting the Result

The negative result indicates that the area under the curve is below the x-axis between the limits. The magnitude of the result represents the total area, regardless of direction.