How to prioritize when calculating the limit of an integral?

[Camilla]

Hey,

So, the original task was as follows: "Calculate the following generalized integral, or prove that it´s divergent.", and the integral itself was:

.

To define this indefinite integral I used limes x---> infinity, and changed the interval of the integral to 0,x.
After that, I factorized x^4+1 and used partial fraction decomposition, which led me to:

limes (x--->infinity):

And this I know I can split into two limits, with one fraction of the integral each. How to proceed henceforth is where I´m lost, though. Do you calculate the integral firstly, and limes secondly? Or the other way around? Because lim(x--->infinty) of both the integrals is obviously 0, and the primitive function of 0 is just an unknown number C, right? And if I have to calculate the integrals before the limit value, how would I do that?

Really grateful for answers!

 

[stapel]

"Calculate the following generalized integral, or prove that it´s divergent.", and the integral itself was:

. . . . .\(\displaystyle \displaystyle \int_0^{\infty}\,\) \(\displaystyle \dfrac{x^2\, +\, 1}{x^4\, +\, 1}\,dx\)

To define this indefinite integral I used limes x---> infinity, and changed the interval of the integral to 0,x.
After that, I factorized x^4+1 and used partial fraction decomposition, which led me to:

limes (x--->infinity): http://www5b.wolframalpha.com/Calcu...16g3cf?MSPStoreType=image/gif&s=8&w=281.&h=50.

I'm sorry, but the link from Wolfram isn't showing up. Kindly please reply with corrections. Thank you!