1 / (x(1+4x^2)) the same as ( 4x^3 + x )^-1 I am in class right now (so cannot go into too much depth here) , and you may know this already, but this problem will involve the natural logarithm (ex. integral 1/x = ln(x) )
You should get something like NOT EXACTLY this:
A ln(x) + B ln( Cx^2 term + D) where A,B,C are constants (that you will fill out!)
Note that any plus could essentially be a minus (by virtue of + negNumber)
and that A B C or D could be 0 (or pos or neg).
Determine which method is the most appropriate: integration by parts, "normal" substitution, trig substitution (I am not ruling anything out, you can get this), integration by parts, inverse chain rule, the fundamental Thm of Calc, etc. etc. It may be even simpler.
Scroll below if you want to see my NON-worked out solution in case you simply want to check your answer when you are confident:
ln(x) - (1/2 (ln(4x^2 + 1))) and no constant in your solution since you are only asked for the antideriv, which means indefinite integral form.