How do you integrate #dx/ sqrt(x^2 - a^2)#?
I takes a couple of substitutions. See explanation.
Use a trigonometric substitution:
With a bit of work you can simplify
If you know this integral, you can skip the next section.
To get that integral multiply by
This gets us:
And the numerator is the derivative of the denominator so we get an
Now that we have integrated the secant, note that due to the first substitution,
Our trigonometry then gets us
So our answer is:
We can rewrite in several ways. Perhaps the simplest is to write:
# = ln((x+sqrt(x^2-a^2))/a)+C#
# = ln(x+sqrt(x^2-a^2)) - lna+C#
But assuming that
Checking the answer by differentiating is left as an exercise. (It's not very tedious.)