Question

Calculus 2 help please. How do know solve an improper integral to find out if it is convergent or divergent? and ho to know?

Answer 1

\[\int\limits_{1}^{+infinity}\frac{ x^2 }{ \sqrt{2x^4+1} }dx\] does it converge? Thanks

Answer 2

if we factor x^4 out of the radical, and simplify we get
\[ \frac{1}{\sqrt{2+\frac{1}{x^4} }} \]
in the limit that is 1/sqrt(2) = 0.707
so in the limit you are integrating
\[ 0.707 \int_{\text{big number}}^\infty \ dx \]

Answer 3

How would you factor out? Sorry rusty algebra

Answer 4

phi's using the fact that for large \(x\),
\[\frac{x^2}{\sqrt{2x^4+1}}\approx\frac{x^2}{\sqrt{2x^4}}=\frac{x^2}{\sqrt2x^2}=\frac{1}{\sqrt2}\]