How do you integrate #y=(1+ln5x)/(x/2)# using the quotient rule?

To integrate the function y=(1+ln5x)/(x/2) using the quotient rule, first, differentiate the numerator and denominator separately. Then apply the quotient rule, which states that the derivative of a quotient of two functions is equal to (the derivative of the numerator times the denominator minus the numerator times the derivative of the denominator) divided by the square of the denominator.

Here's the step-by-step process:

1. Differentiate the numerator: dy/dx = d(1 + ln(5x))/dx = (0 + 1/(5x) * 5) = 1/x

2. Differentiate the denominator: dy/dx = d(x/2)/dx = (1/2) * (d(x)/dx) = 1/2

3. Apply the quotient rule: dy/dx = (1/x * (x/2) - (1 + ln(5x)) * (1/2)) / (x/2)^2

4. Simplify: dy/dx = (1/2x - (1 + ln(5x))/2) / (x^2/4)

5. Further simplify: dy/dx = (2 - (1 + ln(5x))x) / (x^2)

Therefore, the integral of y=(1+ln5x)/(x/2) using the quotient rule is: ∫(1+ln5x)/(x/2) dx = (2 - (1 + ln(5x))x) / (x^2) + C