In Exercises 69–80, determine whether the improper integral co...

Question

In Exercises 69–80, determine whether the improper integral converges or diverges. If it converges, evaluate the integral.

∫₂^∞ (1 / (x√x)) dx

Accepted Answer

Hello there, today we are going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use. Determine whether the improper integral converges or diverges and find its value if it converges. The integral evaluated from 5 to infinity of 1 divided by x 2 multiplied by the square root of x dx. OK, so it appears for this problem we are given an improper integral and we're asked to determine if this improper integral will converge or diverge, and we're asked to determine if it does converge, we're asked to determine what its value of convergence is. So ultimately we are taking this integral and we're determining whether or not it converges or diverges. So now that we know what we're ultimately trying to solve for. Based on our prior knowledge, our first step is, we need to rewrite the integrand. So we need to take 1 divided by x 2 multiplied by the square root of x, which is equal to 1 divided by x 2 multiplied by x of 1 divided by 2, which will be. Equal to 1 divided by x to the power of 5 divided by 2, which will be simply equal to x the power of -5 divided by 2. So that will mean we can write our improper integral as the integral evaluated from 5 to infinity of x to the power of -5 divided by 2. Dx which will. Be equal to the limit as B approaches infinity of the integral evaluated from 5 to B of x to the power of -5 divided by 2 dx. We now need to evaluate our integral. So we need to take the integral, evaluate from 5 to B of x to the power of -5 divided by 2, the x, which will be equal to x the power of 3 divided by 2, divided by 3 divided by 2, and we need to evaluate this at 5B. Which will be equal to -2 divided by 3 multiplied by x to the power of 3 divided by 2, and we need to evaluate this at 5B. And this will be equal to -2 divided by 3 X multiplied by the square root of X evaluated at 5B. Which will be equal to -2 divided by 3 multiplied by B multiplied by the square root of B. Minus parentheses of 2 divided by 3 multiplied by 5 multiplied by the square root of 5. Which will be all equal to, so this will be all equal to 2 divided by 15 multiplied by the square root of 5 minus 2 divided by 3 multiplied by B multiplied by the square root of B. Fantastic. So now we need to find our limit. So when you take the limit as B approaches infinity of parentheses of 2 divided by 15 multiplied by the square root of 5 minus 2 divided by 3 multiplied by B multiplied by the square root of b. Which will be equal to square brackets of 2 divided by 15 multiplied by the square root of 5 minus 2 divided by 3 multiplied by infinity multiplied by the square root of infinity, which will be equal to 2 divided by 15 multiplied by the square root of 5. Which will be equal to 2 multiplied by the square root of 5 divided by 75. So that means without a doubt, the improper integral converges to 2 multiplied by the square root of 5 divided by 75. So that's it, we've officially solved for this problem. We have determined that it converges. Hooray, we did it. So looking at the prom itself, we can safely say once again, without a doubt, our final answer is converges to 2 multiplied by the square root of 5 divided by 75, and that's it. That is our final answer. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

In Exercises 69–80, determine whether the improper integral converges or diverges. If it converges, evaluate the integral.∫₂^∞ (1 / (x√x)) dx

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In Exercises 69–80, determine whether the improper integral converges or diverges. If it converges, evaluate the integral.
∫₂^∞ (1 / (x√x)) dx