Question

Just need someone to check my answer on an integration problem ...
Integral of 1/(1+4x^2) dx = 1/4 arctan(4x)+C? Or am I completely off?

Answer 1

\[\large \int\limits \frac{1}{1+4x^2}dx\]

Hmm I think you're very close. You just forgot that your 4's get squared when you make a substitution. So you don't want to be substituting 1/4. You want a 1/2.
(Assuming that's the route you took toward solving this).

\[\large x=\frac{1}{2}\tan \theta\]\[\large \int\limits\limits\limits \frac{1}{1+4x^2}dx \qquad = \qquad \int\limits\limits\limits \frac{1}{1+4(\frac{1}{2}\tan \theta)^2}dx \qquad = \qquad \int\limits\limits\limits \frac{1}{1+\tan^2 \theta}dx\]

Answer 2

\[\large \color{royalblue}{dx=\frac{1}{2}\sec^2\theta \;d \theta}\]

\[\large \int\limits\frac{1}{1+\tan^2 \theta}\color{royalblue}{dx}\]

Answer 3

I'm a little confused about the substitution part. How did you get that x = 1/2 tan x?

Answer 4

We see a form that looks similar to this in the denominator.\[\large a^2+x^2\]
When we see this form, we make the substitution, \[\large x=a \tan \theta\]\[\large a^2+x^2 \qquad \rightarrow \qquad a^2+(a \tan \theta)^2 \qquad \rightarrow \qquad a^2(1+ \tan^2 \theta)\]Using our trig identity, this will simplify to,\[\large a^2(\sec^2 \theta)\]


In the problem we have here, it appears our a value not where we would like it to be. We have something looking a little more like this,\[\large 1+a^2x^2\]

We will make the substitution, \[\large x=\frac{1}{a}\tan \theta\]That will get rid of the a value for us.\[\large 1+a^2x^2 \qquad \rightarrow \qquad 1+a^2\left(\frac{1}{a}\tan \theta\right)^2 \qquad \rightarrow \qquad 1+\tan^2 \theta\]\[\large = \sec^2\theta\]

Answer 5

If you've learned this using a different method, such as based off of memorization, then this might be a lil confusing D:

Answer 6

Our goal is to get rid of the addition or subtraction that may be in the denominator by using a Trig Identity. It will make the integral much much easier to deal with.

Answer 7

no i appreciate your explanation! I have to run now, but I think I can make sense of this now. :) If I get terribly confused trying to understand it, I'll mention you in a reply later on. :D

Answer 8

yeah that makes sense :)

Answer 9

kk c: have nice day

Answer 10

you too! Thanks!