Question

Evaluate the integral 4(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 4(x-a)(x-b) dx, we can expand the expression and then integrate term by term. The final integral is (4/3)x³ - 2(ax + bx) + 2abx + C, where C is the constant of integration.

Step-by-step explanation:

To evaluate the integral 4(x-a)(x-b) dx, we can expand the expression and then integrate term by term. Using the distributive property, we have 4(x-a)(x-b) = 4(x² - ax - bx + ab) = 4x² - 4ax - 4bx + 4ab. Now, we can integrate each term separately.

1. Integrating 4x² gives us (4/3)x³ + C₁, where C₁ is the constant of integration.
2. Integrating -4ax gives us (-2ax²) + C₂, where C₂ is the constant of integration.
3. Integrating -4bx gives us (-2bx²) + C₃, where C₃ is the constant of integration.
4. Integrating 4ab gives us (2abx) + C₄, where C₄ is the constant of integration.

Combining all the terms, the final integral is (4/3)x³ - 2(ax + bx) + 2abx + C, where C is the constant of integration.