What is the method for evaluating integrals with linear terms?

[maobadi]

Evaluate the integral

 

[tiny-tim]

Hi maobadi!

Split the limits of integration into two or more intervals on each of which the integrand has the same sign, so that, on each interval, you can replace || by either +() or -()

 

[HallsofIvy]

Since the quantity inside the absolute value sign is a product of linear terms, x(x-5)(x+7), it is easy to see that it is 0 exactly at x= -7, x= 0, and x= 5. Break the integral into two parts, from -7 to 0 and from 0 to 5, do those integrals separately (being careful about the sign) and add the results.