Final answer:
The integral of x^2(x^3 + 20) dx, after the substitution u = x^3 + 20, becomes the integral of (1/3)u^(1/2) du. This evaluates to (2/3)u^(3/2), which simplifies the original problem.
Step-by-step explanation:
The given information suggests a problem involving substitution in integration, a fundamental concept in calculus. You have a function of x, and you are making a substitution to simplify the integral. The student is looking to rewrite the integral x2x3 + 20 dx in terms of a new variable u, which is equal to x3 + 20. After this substitution, the differential dx will be expressed in terms of du, and the integral becomes the integral of u1/2 with respect to u. The integration process will involve using the power rule.
By integrating 1/3 u1/2 du, we obtain 2/3 u3/2 as the antiderivative. Since u = x3 + 20, the final answer in terms of u will be 2/3 u3/2.