Evaluate the integral log(x+1) - log x dx/x(x+1).

To evaluate the integral value of log (x+1) - log x dx/x (x+1), we will use substutitution method.

Answer: ∫ log (x + 1) - log x dx/x (x + 1) = -1/2(log (x + 1))² - 1/2 (log x)² - log (x + 1) log x + C

Here is a detailed explanation.

Explanation:

Let log (x+1) - log x = t.

On differentiating the above equation with respect to 'x', we get

1/(x+1) - 1/x = dt/dx

d/dx log x = 1/x

On solving this we'll get,

-1/x (x+1) = dt/dx

dx = -x (x+1)dt

On substituting the value t and dt in the equation,we get

∫ t/x (x+1) - x (x+1) dt

On solving this we get

∫ -t dt = -t²/2 + C

On substituting the values of 't' in the equation, we get

-1/2 [log (x+1) - log x]²+ C

-1/2(log (x+1))² - 1/2 (log x)² - log (x+1) log x + C