cosh^{-1}(x) has for its derivative 1/√[x^{2}-1], so if we get something that looks similar in form (by a u-substitution, for example), we can integrate.

1/√[x^{2}-5] is almost there, but we need to do something about the 5 (based on the derivative, we need a 1 there). If we factor out the 5 from the quadratic, we can recover the 1:

1/√[5(x^{2}/5-1)]

And in fact, we can take √5 out of the whole denominator now. Because it's just a constant, we can take it out of the entire integral:

(1/√5)*Int(1/√[x^{2}/5-1] dx)

Next we want to pull the x^{2}/5 together into a single term squared:

x^{2}/5 = (x/√5)^{2}

This is important because now we can make a simple u-sub and get rid of it entirely:

u=x/√5

√5 du=dx

The integral is transformed into

dx/√[ (x/√5)^{2}-1 ] ==> (√5 du)/√[u^{2}-1]

The √5 we pulled out earlier (in the denominator) and the √5 we created with the u-substitution (in the numerator) cancel each other out. The integral we're left with will exactly give us cosh^{-1}(u)=cosh^{-1}(x/√5).

When you evaluate cosh^{-1} at the two limits of integration, you can substitute in the ln(...) definition for evaluation instead (since they differ by a constant, the constant will disappear when you subtract). Just remember to carry over the x/√5 when you go from cosh^{-1} to ln