Help with anti differentiating

[Lols123]

Hey guys

just wondering if you guys knew a shortcut to anti diffing tan equations such as 3/(9+4x^2)

[sajib_mostofa]

Firstly, I'd want to make a substitution so that I can get it in the form , which will be easy to integrate.

So in this case, let , you end up with .

Then take the 3/9 out of the integration sign and integrate it normally.

Or you could try and memorise the general tan form that you obtain when you integrate something of the form but I wouldnt advise it.

[Lols123]

hmm am i missing a step
take this one instead: dx / (8 + 2x^2)

I took out half from the equation
1/2 (sigma) 1 / (4 + x^2)

and from there get 1/2 tan^-1 (x/2)
but the answer says 1/4 instead

D:

[sajib_mostofa]

so if you have , you let .

You end up with . Factor our the 4 and hence, take out 1/4 out of the integrand sign.

[b^3]

could also do it this way if you want
remeber the antiderivative is
so when you take it out you need  to keep a 2 on the top so put it there and times the front by 1/2 again
so that you get
now you get the required answer