Some time ago I came across to the following integral: I=∫011−x1+xdxlnx What are the hints...
Some time ago I came across to the following integral:
What are the hints on how to compute this integral?
Answer & Explanation
Make the substitution and do a little algebra to get the value of the integral to be
Now Taylor expand the denominator and get
If we can reverse the order of sum and integral, we get
The integral inside may be evaluated exactly, and the result is the sum
I tried with Differentiation under integration sign:
Then as usual,
I used Mathematica to evaluate (it gives difference of two Harmonic numbers) and its integral. The integral is just . Note that and putting , the result follows.
Differentiating (1) with respect yields
Now consider polygamma function
Hence by using (4) we obtain
Using (3) and (5) then (2) becomes