Help with really difficult calculus BC question?

Question

I'm stuck on a problem for BC calc, and I don't think my logic is sound...could anyone look over this and see if it makes sense at all?

Problem: Show that the improper integral: 0 ~ infinity (e^-x / square root x ) dx converges.

I broke it up into two parts:

Integral of 0 ~ infinity (1 / e^x) dx converges to 1,

Integral of 0~ infinity (1 / square root x) diverges to infinity.

So I said that since the integral e^-x converges to 1 and the integral of 1 / square root x diverges to infinity, the result was 1 / infinity, which would be 0, thus converging the original function to 0.

Does this make sense at all? It seems odd that an integral would reach 0...

Answer 1

Your approach doesn't work, because the 1 and infinity that you calculated should be MULTIPLIED together, not divided.

I'm thinking that maybe you apply l'Hospital's rule to determine that the integrand goes to 0. But that doesn't prove that the integral converages. (For example, x^-1 goes to 0 as x goes to infinity, but the integral of x^-1 (from 0 to infinity) does not converge.

You can integrate by parts if you define u as e^-x and define v as 2 x^.5, but it still produces an integral (of v du) that I don't know how to integrate.

Hope that's useful to you.
Sorry I don't have an actual solution for you.

Answer 2

AP calculus AB is relating to the equivalent of a school Calc a million direction. AP calculus BC is relating to the equivalent of a school Calc 2 direction. on account which you're asking approximately BC, could i assume which you have taken AB? if so, the homework load and consider out project is approximately what you're conscious of. in case you haven't any longer taken AB first or a minimum of a calculus direction of a few type, you could. as quickly as I even have taught AP Calc, my scholars ought to anticipate some million hour of math homework on an well-known basis (with approximately 20 minutes in college time to start). The suggestions are of direction extra durable, yet once you have understood the maths you have taken so some distance, it may well be a organic progression. i did no longer assign any outdoors initiatives. tests have been complicated however the purpose replaced into to coach scholars for the AP examination.

Answer 3

Your mistake is that in general, for two functions f and g, the integral of f/g is different from integral of f / integral of g.

To prove that the integral converges, you need to split it between 0 and 1 and between 1 and infinity:
- between 0 and 1: exp(-x)/sqrt(x) <= sqrt(x) and the integral of sqrt(x) converges at 0.
- between 1 and infinity: exp(-x)/sqrt(x) <= exp(-x) and the integral of exp(-x) converges at infinity