Final answer:
To evaluate the definite integral, we expand and simplify the expression, integrate each term separately, and add the constant of integration.
Step-by-step explanation:
To evaluate the definite integral ∫49(30−x3/2)2x dx, we can expand the expression and simplify it before integrating:
∫49(30−x3/2)2x dx = ∫49(900 - 60x3/2 + x3)dx = ∫45000 - 2940x3/2 + 49x3 dx
Next, we can integrate each term separately:
∫45000 - 2940x3/2 + 49x3 dx = 45000x - 2940 * (2/5)x5/2 + 49 * (1/4)x4 + C
where C is the constant of integration. So, the definite integral is:
∫49(30−x3/2)2x dx = 45000x - 588x5/2 + 12.25x4 + C