I have seen plenty of proofs of the derivative forms of arcsin, arctan, arcsec... However, I would like a proof of how to go from $$\int\frac{1}{\sqrt{x^2 +4}}dx $$
WITHOUT memorizing that info in the table.
I think the best way to do it is via drawing the triangle, but I don't know how to do it, and I can't find a proof anywhere.
You can first use the substitution $x=2\tan \theta$, and simplify, the integral result becomes $$ \sec\theta +c=\frac{\sqrt{x^2+4}}{2}+c$$ (HINT: To get its value just draw a a right angle triangle with sides $x$ and $2$) If the integral is without square root the result will be$$\frac{1}{2} \theta+c=\frac{1}{2} \arctan \frac{x}{2}+c$$ If you face any problem please just let know