How do I solve the integral of dx/(3x^2+1)?

Question

zepdrix · Answer

Mmm looks like some type of arctangent.
Have you learned about trig substitutions?

domebotnos · Answer

Yup, pretty sure it has something to do with arctan. arctan(sqrt(3)x)?
It looks right to me but wolfram says it's wrong so I think I'm missing something.

zepdrix · Answer

You can bring the 3 into the square,
$$\Large\rm \int\limits \frac{1}{\color{orangered}{\left(\sqrt3 x\right)^2}+1}dx$$Mmmmm yah looks like you've got the right idea. lemme think...

zepdrix · Answer

Oh we're missing a factor of sqrt3 on the top.

zepdrix · Answer

If you do a u-sub it might make more sense,$$\Large\rm u=\sqrt3 x, \qquad\qquad\qquad du=\sqrt3dx\qquad\qquad\qquad \frac{1}{\sqrt3}du=dx$$See where the 1/sqrt3 is coming from?$$\Large\rm \frac{1}{\sqrt3}\int\limits \frac{1}{u^2+1}du$$

domebotnos · Answer

Yes, so that'll equal 1/sqrt3 arctan(sqrt(3)x), right? That looks correct

zepdrix · Answer

ya looks good :)

domebotnos · Answer

cheers