# What is the integral of #int 1 / (sqrt(x+1) - sqrt(x)) #?

##### 1 Answer

#### Explanation:

When I first looked at this one I thought it would be quite difficult, but in fact all we have to do is remember a basic technique from algebra: conjugate multiplication.

You may recall that the conjugate of a number is what you get when you switch the sign in the middle of a binomial. For example, the conjugate of

We can see that even though our original expression contained square roots, our answer didn't - which, as we will see, is very useful.

In the denominator of our integral, we have

Doing some conjugate multiplication in the denominator,

The middle terms cancel out, leaving us with

And then the

We see the beauty of conjugate multiplication now, as our seemingly complicated integral is now reduced to

Using the properties of integrals,

And because

Now all we have is a case of the reverse power rule in both integrals, making them simplify to

And that, ladies and gentlemen, is our final answer.