Question

Evaluate the integral ∫2(x-a)(x-b)dx. (Assume a ≠ b. Remember to use absolute values where appropriate. Use c for the constant of integration.)

Answer 1

To evaluate the integral ∫2(x-a)(x-b)dx, expand the expression and use the power rule of integration. The result will be a quadratic polynomial.

Step-by-step explanation:

To evaluate the integral ∫2(x-a)(x-b)dx, we can expand the expression and then use the power rule of integration. The result will be a quadratic polynomial.

Expanding the expression, we have: 2(x-a)(x-b) = 2(x² - (a+b)x + ab)

Now, we can integrate each term separately: ∫2(x² - (a+b)x + ab)dx = ∫2x²dx - ∫2(a+b)xdx + ∫2abdx

Using the power rule, the integral of x² is (2/3)x³, the integral of (a+b)x is (a+b)(x²/2), and the integral of ab is 2abx. Adding these results together, we get: (2/3)x³ - (a+b)(x²/2) + 2abx + C, where C is the constant of integration.