The discussion revolves around determining the convergence of the integral of the function 1/sqrt(x^4+x^2+1) from 1 to infinity. Participants are exploring methods to evaluate the integral and discussing the implications of certain transformations on its convergence.
There is an ongoing exploration of different approaches to evaluate the integral. Some participants have provided guidance on transformations and substitutions, while others are questioning the necessity of integrating the exact expression. The discussion reflects a mix of interpretations and methods being considered without a clear consensus on the best approach.
Participants are working within the constraints of homework rules, focusing on understanding the convergence of the integral rather than finding a complete solution. There is mention of specific terms in the integral that may affect convergence, which are under discussion.
Don't forget the dx.Neil21 said:Ok, so I complete the square and got \int \frac{1}{((x^{2}+\frac{1}{2})^{2}+\frac{3}{4})^{\frac{1}{2}}} . I am still not seeing a way find an antiderivative though.
What I first saw was rewriting the expression as $${\frac{1}{(x^4+x^2+1)^{1/2}}}={\frac{1}{((x^2+({\frac{1}{2}))^2}+(\frac{\sqrt{3}}{2})^2)^{1/2}}}$$.Mark44 said:Don't forget the dx.
$$\int \frac{dx}{((x^{2}+\frac{1}{2})^{2}+\frac{3}{4})^{\frac{1}{2}}}$$
I haven't worked this through, yet, but here's what I would do, FWIW.
Let u = x2 + 1/2
So ##x = \sqrt{u - 1/2}##
Then du = 2xdx
So ##dx = \frac{du}{2x} = \frac{du}{2\sqrt{u - 1/2}}##
After making the substitution and getting everything in terms of u and du, I would try for a trig substitution. I think that might work.
Ooops. Missed that. Sorry guys.Mark44 said:@chet, I think he already established that in the opening post.