💬 👋 We’re always here. Join our Discord to connect with other students 24/7, any time, night or day.Join Here!

Like

Report

WZ
Numerade Educator

Like

Report

Problem 55 Hard Difficulty

The integral
$$ \int_0^\infty \frac{1}{\sqrt{x} (1 + x)}\ dx $$
is improper for two reasons: The interval $ [0, \infty) $ is infinite and the integrand has an infinite discontinuity at 0. Evaluate it by expressing it as a sum of improper integrals of Type 2 and Type 1 as follows:
$$ \int_0^\infty \frac{1}{\sqrt{x} (1 + x)}\ dx = \int_0^1 \frac{1}{\sqrt{x} (1 + x)}\ dx + \int_1^\infty \frac{1}{\sqrt{x} (1 + x)}\ dx $$

Answer

$\pi$

Discussion

You must be signed in to discuss.

Video Transcript

the problem is anti girl from zero to infinity one over Richard Wax on one pass axe yet is improper For two reasons Interval zero to infinity is infinite on his immigrant has on infinity discontinuity at zero you're validate by this president as Osama mean proper into gross off top two and one as follows before we compute thiss into grow First, let's compute infill in indefinite integral of this function one over motive acts one of us. Yes. So here we use you substitution that you is equal to blue tive acts. Then you squire is he going to ice Ah nde to you, Do you? It's Niko too. Yes, this indefinite Integral is legal too. And to grow till you you over you tasks one plus use Claire, this's the cultures into grow to see you over Juan us you square. So this is you Go to two hams, our tenant. You You is brutal acts. This is two towns tenant U. Until facts, some cast the numbers. See now and girl from zero to one one over one plus x with axe. Yes. As the cultures limit A goes to zero on DH and integral from a one of dysfunction one or one plus x. I'm saluting X. Yes. Since they're going to the limit, A goes to zero. Andi, this is equal to act two times back turned util acts from a to one. This is a cult, you lam it. Hey, goes to zero two hams back ten and one, Linus, two hams act and beautiful, eh? But make us two zero. A tenant zero is zero. So justice too shoot terms. Our tenant Juan severally was integral from one too pointy. One over one plus acts Primitive. Yes, this's the cultures that limit some B goes to infinity of the function two times can it on beam minus two times act tenant one The insolent biggles to infinity. Two hams A tenant B goes to two times high over to So the answer is equal to high minus two hams. Cannon one. Since this one class is one plus one, is he going to chew? Tam's attendant one plus five minus two hams attended one. This number plus this number. But this is for two. Hi. This is a wider