is sinh^-1(x+2/2) equivalent to 1/2ln|x+2| +c

Question

The questions is evaluate the integral dx/ Sqrt x^2+ 4x + 8

my answer was 1/2ln|x+2| +c , but the books answer is sinh^-1 (x+2/2) +c

can they both be correct?

Answer 1

The books answer is correct :(

One way to check is to graph both solutions, for example with a graphing calculator, or PC program. If they look nothing like each other then obviously one is wrong!

You had, 1/sqrt(x^2 + 4x + 8)

Or, 1/sqrt{(x+2)^2 + 4}

Or, 1/sqrt(u^2 + a^2), where u = x + 2 and a = 2.

That last expression, 1/sqrt(u^2 + a^2), is a standard integral

and evaluates as sinh^-1(u/a) + c.