# How to calculate this? #int_0^1 log(1-x)/xdx#

##### 1 Answer

See below.

#### Explanation:

Unfortunately the function inside the integral will not integrate to something that cannot be expressed in terms of elementary functions. You will have to use numerical methods to do this.

I can show you how to use a series expansion to get an **approximate value**.

Begin with the geometric series:

Now integrate with respect to

Integrating the left hand side:

Now integrate the right hand side by integrating term by term:

So it follows that:

Now divide by

So we now have power series expression for the function we originally started with. Finally, we can integrate again to get:

Integrating the right hand term by term side gives us:

Evaluating the limits to four terms will give us an approximate value:

Now, this is only to four terms. If you would like a more accurate number simply use more terms in the series. For example, going to the 100th term:

As an aside, if you work through the exact same process but use summation notation (i.e. with big sigma rather than writing out the terms of the series) you will find that:

which is just the Riemann-Zeta function of 2, i.e:

We actually already know the value of this to be:

Hence the exact value of the integral can be deduced to be: