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A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
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
Thus, the integral value of log (x+1) - log x dx/x (x+1) is -1/2(log (x+1))2 - 1/2 (log x)2 - log (x+1) log x + C
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