dividing definite integrals

Question

∫(top: b, bottom: a) ∫(f(x)/g(x))dx=(top: b, bottom: a)∫f(x)dx÷(top: b, bottom: a)∫g(x)dx

 

true or false?

Answer 1

∫[f(x) / g(x)] dx = ∫f(x)dx  / ∫g(x)dx

 

 

This is not true.  On the left side of the equation, you are taking the integral of one quotient.  On the right side of the equation, you taking quotient of two integrals.  If you were to define values for f(x) and g(x), the results of the left and right side of the equation will be different from each other.