In Exercises 35–68, use integration, the Direct Comparison Tes...

Question

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.

∫ from 2 to ∞ of (dx / √(x² - 1))

Accepted Answer

Welcome back everyone. Determine whether the improper integral from 3 up to infinity of d x divided by the square root of x2 minus 4 converges or diverges. A says converges and B says diverges. So for this problem, let's remember that if we want to evaluate an improper integral, we basically can begin by rewriting it as a limit. So integral from 3 up to infinity of 1 divided by a square root of x2 minus 2 d x. Can be written as a limit. As B approaches infinity, that's because we have an upper limit of integration. Equal to infinity or basically approaching infinity, right? And we're going to take the limit of the integral from 3 up to B, so we're replacing infinity with B. And the rest remains the same, so we're going to integrate 1 divided by the square root of x2 minus 4 dx. And now let's focus on that integral. We are also going to label the limit as L. And let's evaluate the integral from 3 up to some value B. We know that B approaches infinity. And we are going to integrate one divided by. Square root of X2 minus 4 d x. Now notice that we can rewrite 4s 22, which essentially implies that we can use the following formula to evaluate the integral. Remember that the integral of dx divided by a square root of x2 minus a 2. is going to be equal to ln of the absolute value of x plus square root of x2 minus a 2 plus a constant of integration c. So in this case we're going to have ln of the absolute value of x plus square root of x2 minus a 2, right? And basically we know that a is 2 and a 2 is basically 4, so we're going to subtract 4. And we went to evaluate the integral from 3 up to B. So now Let's go ahead and apply the fundamental theorem of calculus part. 2 That would be ln of the absolute value of. B + square root of. B2 minus 4 Minus ln of the absolute value of 3 + square root of 32 minus 4. Now we can use the properties of. Logarithms to rewrite it as ln of the absolute value of b plus square root of. B 2 minus 4 divided by 3 plus square root of 9 minus 4 gives us 5, right? And now let's go back to our limit. The limit l is going to be equal to what we're going to do is write brackets showing that we're trying to evaluate. Our function as b approaches infinity. Remember that we cannot substitute infinity directly. So we're basically going to write infinity instead of b, but this is not really accurate, and that's why we're using brackets to show that we are basically visualizing what's happening when b approaches infinity. So that blm of the absolute value of. Infinity plus square root of infinity 2 minus 4 basically gives us infinity divided by 3 + square root of 5. So in the numerator we have infinity plus square root of infinity. This part approaches infinity, and infinity divided by a constant is still infinity. Notice that our constant is positive, right? And we can drop the absolute value because we get. Positive infinity. Now ln of infinity is also infinity, right? The natural logarithm of an infinitely large number. Gives us an infinitely large number, so l is equal to infinity and because Our limit is equal to infinity, not some finite value. We can conclude that the integral diverges. So the correct answer for this problem corresponds to the answer choice B. Thank you for watching.

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.∫ from 2 to ∞ of (dx / √(x² - 1))

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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.
∫ from 2 to ∞ of (dx / √(x² - 1))