I need to solve this integral:
$$\int{\frac{\mathrm dx}{\sqrt{1+x^2}}}$$
First I thought it was easy, so I tried integration by parts with $g(x)=x$ and $g'(x)=1$:
$$\int{ \frac{x^2}{(1+x^2)^{\frac{3}{2}} }}\,\mathrm dx $$
But I've made it even more complicated than before, and if I want to solve it again by parts I'll have $g(x)= \frac{x^3}{3}$ , and I will never end integrating.
How should I solve it?
Edit
Trying this way: $x= tg(t)$, then I get:
$$ \int{ \frac{1+tg^2(t)}{ \sqrt{1+ tg^2(t)} } dt}= \int{ \sqrt{ 1 + tg^2(t) } dt } $$
But it doesn't remind me anything, I still can't solve it.
Best Answer
When you see $1+x^2$ you should without hesitation think "Let $x=\tan\theta$."
If you're comfortable hyperbolic trig functions, then metacompactness's suggestion is better. :)