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Numerade Educator

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Problem 67 Hard Difficulty

In Exercises $35-68$ , use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.
$$\int_{-\infty}^{\infty} \frac{d x}{\sqrt{x^{4}+1}}$$

Answer

converges.

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Video Transcript

he is clear is that when you married hearing so we could use the limit comparison test with G of X equal to one over X square plus one. We're going to use the limit comparison tests as X approaches infinity for one over a square root of X to the fourth plus one over one over X Square plus one, which is equal to the limit as X approaches infinity for X square plus one over square root of X to the fourth plus one, which is equal to the limit as X approaches infinity for the square root of extra fourth plus two X Square plus one over X to the fourth plus one, which is equal to one. So since it's non mon zero for night number, it will either both converge or diverge. So we get from negative infinity to infinity. For D X over X square plus one, this is equal to the integral from negative infinity to zero for D X over X square plus one plus the interval From there oh, to infinity for D X over explore plus one. We know that G of X is even so. This can be re written as two come zero to infinity burn D over X over X square plus one which is able to to limit of a approaches Infinity there Oh, to a theatrics over at square pulse one, which is equal to two limit of the approaches infinity for inverse tangent from zero to a which is equal to two times pi house over minus zero is equal to pine. So this converges.