How can trigonometric substitution help solve this integral?

[colorado]

integrate (x^2) / (4+x^2)^(3/2)

Im not allowed to apply hyperbolic functions to this and have been trying to solve applying to a 90 deg. angle.

x = 2tan(theta)
x^2 = 4tan^2(theta)
dx = 2 sec^2(theta)

Hopefully you can se where I am going with this (trigonomic substitution)

Im ready to give up!

 

[snipez90]

Hmm that substitution worked for me. I think a similar (or the same) problem appears in Apostol. Anyways try the substitution again. Check your arithmetic and remember that 1+tan^2(x) = sec^2(x). Also don't forget the dx part.

 

[colorado]

Im not forgetting about the dx. I have been working on this for 2 hours now, please give me a little more than that. Here is what I have so far that I believe to be right:

(2)integral tan^2(theta) / sec(theta)

 

[rootX]

Im not forgetting about the dx. I have been working on this for 2 hours now, please give me a little more than that. Here is what I have so far that I believe to be right:

(2)integral tan^2(theta) / sec(theta)

Remember his said thing
tan^2 = 1 + sec^2
Use integration table for finding integral of sec t

 

[snipez90]

Oh, you've gotten to that part. Well changing everything to sin and cos you get sin^2(theta) / cos(theta). At this point I think it's easier if you write it in terms of tan(theta)*sin(theta) and then try integrating by parts. I chose u to be tan(theta) and dv = sin(theta)d(theta). Either way you'll have to integrate a somewhat obscure but really rather well-known trig expression.

EDIT, or rootX found an easier way to manipulate that and integrate.

 

[snipez90]

Wait nevermind, I was wondering why the integration by parts came out so nicely. If you integrated by parts like I did it is equivalent to just convering sin^2(theta) to 1- cos^2(theta) and dividing by cos(theta) you get sec(theta) - cos(theta) so again it comes down to the antiderivative of sec(theta) which you could look up. The derivation requires an insight.

 

[tiny-tim]

tan^2(theta) / sec(theta)

= (sec^2 - 1)/sec = sec - cos

 

[colorado]

okay, so now I've got:
(2) integral sec(t) - cos(t)

Do I go ahead and take the integral at this pont? the integral of sec(t) involves (ln) and I don't believe that to be correct.

 

[tiny-tim]

okay, so now I've got:
(2) integral sec(t) - cos(t)

Do I go ahead and take the integral at this pont?

Yes! Why not??

the integral of sec(t) involves (ln) and I don't believe that to be correct.

i] why?

ii] try it anyway!

 

[snipez90]

Yeah if I remember the derivation correctly you multiply sec(x) by [sec(x) - tan(x)]/[sec(x) - tan(x)] and note that in the resulting expression, the denominator's derivative is the negative of the expression in the numerator. This suggests an integral involving ln.