[Calculus 2] Solve the integral (question in comments).

Question

anonymous · Answer

If someone could explain how to do this one, that would be awesome. It has me stumped :/.

kainui · Answer

Make the substitution x=tan(theta) and I think it should work out nicely.

kainui · Answer

I can explain it better than that if you'd like me to, but I don't know if you already know trig substitution or just didn't see it or what.

anonymous · Answer

@Kainui If you could go into a bit more detail, that would be awesome. I am QUITE fuzzy on Trig substitution and my final is tomorrow! *facepalm*

anonymous · Answer

A better solution would be to do a u-substitution. 
You can have
$$u = 1 + x^{2}$$ 
$$du = 2x dx$$
$$\frac{ du }{ 2 } = x dx$$
 With this, you can substitute it back into the equation

$$\frac{ 1 }{ 2 }\int\limits_{}^{}\frac{ du }{ u }$$ 

It's easier than doing trig substitution

kainui · Answer

Alright, so basically trig sub plays off of this one, main rule.
$$\sin^2(\theta)+\cos^2(\theta)=1$$ Now, you can multiply both sides by the same thingsee if you divide it through by cosine of theta it becomes:
$$\tan^2(\theta)+1=\sec^2(\theta)$$

Since your final is tomorrow, I'll just continue explaining this if you'd like, even though he has a better way of doing it.

kainui · Answer

You won't always be able to do u-substitution though, even though you lucked out on this one. =P

anonymous · Answer

I agree, in this case, u-substitution worked out very nicely. Though if you could refresh on trig substitution, it will be very helpful for some problems :).

kainui · Answer

It would be nice, and a good way of checking yourself if you did both ways and came up with the same answer.

anonymous · Answer

Ah, thank you both very much! :)