Question

Evaluate \( \int_1^3 [x - 1] \, dx \)

• A. 4
• B. 6
• C. 8
• D. 10

Hint:

When solving integrals with simple polynomials, first find the antiderivative and then evaluate the result using the limits of integration.

Answer 1

The Correct Option is B

The given integral is: \[ \int_1^3 [x - 1] \, dx \] We need to integrate the function \( f(x) = x - 1 \) over the interval \( [1, 3] \). First, compute the antiderivative of \( f(x) = x - 1 \): \[ \int (x - 1) \, dx = \frac{x^2}{2} - x \] Now, evaluate this antiderivative from 1 to 3: \[ \left[ \frac{x^2}{2} - x \right]_1^3 = \left( \frac{3^2}{2} - 3 \right) - \left( \frac{1^2}{2} - 1 \right) \] \[ = \left( \frac{9}{2} - 3 \right) - \left( \frac{1}{2} - 1 \right) \] \[ = \left( \frac{9}{2} - \frac{6}{2} \right) - \left( \frac{1}{2} - \frac{2}{2} \right) \] \[ = \frac{3}{2} + \frac{1}{2} = 6 \] Thus, the value of the integral is 6.