Final answer:
The integral ∫|13x3−125| dx should be solved piece by piece depending on where the expression inside the absolute value is positive or negative. After integrating the expression, we expect the x4 term to have a positive coefficient and for there to be a constant of integration, C. Thus, the correct answer is 4/13x4+125x+C.
Step-by-step explanation:
To evaluate the integral ∫|13x3−125| dx, we first need to understand that the absolute value sign | | will affect the integral depending on the sign of the expression inside it. Let's first consider the inside function 13x3 -125 and find where it is negative and where it is positive. For x values that make the expression positive, we integrate normally, and for x values that make it negative, we integrate its negation.
Looking at the options, options (a), (b), and (c) all include a constant of integration 'C', which tells us that we are looking for an indefinite integral. Option (d) is incorrect because it doesn't include the constant of integration.
To decide between options (a), (b), and (c), we must realize that the integral of a positive function should also be positive, so if the integral of 13x3 -125 is negative, we would need to negate it due to the absolute value, and if it's positive, we can directly integrate. After integrating, we should expect the term with x4 to have a positive coefficient since the polynomial term 13x3 will result in an x4 term after integration, which cannot have a negative coefficient based on the standard rules of integration.
Therefore, the correct integral should have a positive coefficient for the x4 term and should include the integration constant C, which means the correct answer is option (b): 4/13x4+125x+C.