Which of the improper integrals in Exercises 63–68 converge an...

Question

Which of the improper integrals in Exercises 63–68 converge and which diverge?

∫ from 6 to ∞ of (1 / √(θ² + 1)) dθ

Accepted Answer

Welcome back everyone. Determine whether the improper integral converges or diverges. Integral from 5 to infinity of 1 d x divided by a square root of x2 + 9. A says converges and b says diverges. So, first of all, let's notice that the radicant in the denominator is always positive, right, because we have x2 + 9 and therefore the integrant is valid from 5 to infinity. We're going to rewrite the improper integral as a limit. So we have the integral from 5 to infinity of d x divided by square root of X2 + 9 We can rewrite it as limit as b approaches infinity of the integral from 5 up to b, so we're basically replacing infinity with b where b approaches infinity, right? D X divided by square root of x2 + 9 and now let's recall that the integral of d x. Divided by square root of X2 + A2 Is equal to. Ln of the absolute value of x plus square root of x2 + a2 plus a constant of integration c right. So now, in this case, Our integral, let's call it I. is going to be equal to the limit as b approaches infinity. Of lm of the absolute value of x plus Square root of x2 + 9, right? Essentially a is equal to. 3, right? Or basically we can just say that a squared is equal to 9. So this is how we get 9 within the radicant of the natural logarithm and what we want to do is evaluate this result from 5 up to B and then we will focus on the limit. So that be limit as b approaches infinity. Off LM Of the absolute value of b plus square root of. B2 + 9 Minus LM. Of the absolute value of 5 plus Square root of 52 + 9 What we can also do is remove the absolute value signs, right? Because between 5 and infinity, these sums are positive. So we're going to remove one of them, and then we're going to remove the 2nd 1. And now let's go ahead and apply the properties of logarithms. We're going to have limit as B approaches infinity. off since we have a difference, we can use the quotient rule. In the numerator, we have b + square root of. B2 + 9 Divided by 5 plus square root of 34. So now notice that as b approaches infinity, the numerator approaches infinity since we have infinity plus square root of infinity divided by a constant. So overall we have an infinitely large number. Taking log of an infinitely large number gives us infinity. So the limit is equal to infinity since it's not a finite number. We can conclude that the integral diverges, and the correct answer is option B. Thank you for watching.

Which of the improper integrals in Exercises 63–68 converge and which diverge?∫ from 6 to ∞ of (1 / √(θ² + 1)) dθ

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Which of the improper integrals in Exercises 63–68 converge and which diverge?
∫ from 6 to ∞ of (1 / √(θ² + 1)) dθ