Final answer:
The integral ∫ (6x)/(x³-b) dx is solved by substitution, leading to the result (1/3) ln|x³-b| + C, where C is the constant of integration.
Step-by-step explanation:
The integral in question is ∫ (6x)/(x³-b) dx. To evaluate this integral, let’s first consider a substitution method given that the derivative of the denominator is linearly related to the numerator. Let u = x³-b, which implies that du = 3x² dx. We need to adjust the numerator accordingly to match with our du, necessitating the multiplication and division by 3. Therefore, we can rewrite the integral as (1/3) ∫ du/u, which is a standard form for the natural logarithm function. Hence, the solution to the integral is (1/3) ln|u| + C, where C is the constant of integration.
Substituting back the value of u, we get (1/3) ln|x³-b| + C as the final answer. This integral does not directly correlate to the other provided information, such as Equation 10.11 or Figure 10.13, which seem to describe different mathematical scenarios unrelated to the given integral problem.