Integral of 1 / (x^2 + 2) dx ?

[bob012345]

The image in post #36 shows why this substitution works. For $x^2 + 2$, the length of the leg adjacent to the angle is $\sqrt 2$ rather than 1.

Of course and the point I made was the method I used works for any $a$. That's why I used $a$ and not $1$. I found it easier than manipulating the integral first to make it a $1$ or using complex numbers.

 

[vela]

Now try it with a hyperbolic trig substitution.

 

[bob012345]

Now try it with a hyperbolic trig substitution.

I tried it and it was not easier by any means.

 

[plum356]

I think that the most straightforward way is to factorise $x^2+2$ into $2\left(1+x^2/2\right)=2\left[1+(x/\sqrt 2)^2\right]$.
When you see an integral close to one that you would usually find in an integration table, $(1+x^2)^{-1}$ for instance, try adding a $0$ or factorising something.

 

[benorin]

You guys have beaten this integral in a few ways, OP did you follow any of them? The easy way involves $1+\left(\tfrac{x}{\sqrt{2}}\right) ^2$ and a trig substitution. You probably haven’t learned series methods yet but that would of course be much harder.

I think you can safely lock this thread.