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Problem 82 Hard Difficulty
Show that if $ a > -1 $ and $ b > a + 1 $, then the following integral is convergent
$$ \int_0^\infty \frac{x^a}{1 + x^b}\ dx $$
Answer
$\lim _{d \rightarrow \infty} \frac{1}{-\epsilon d^{\epsilon}}=0$
Topics
Integration Techniques
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Video Transcript
The problem is show that if its greatest on ninety one and B is created on a past one, then the following into grow is converted. This improper integral we can, right? It's integral. Is he called to integral from zero to one ax to a over one plus X to be the eggs class into girl from one to the infinity of ax to a for one flat sachs to be power? Yes, on the first part is is a definite integral, so it will be a constant a number. So if the second heart is converted, then are integral from zero to infinity of dysfunction is commitment. Now look at this improper integral. We can write disfunction, eyes divided acts to its power for the numerator and the denominator. So this is one over one. Over. Ax to ace Power class Act two I mean minus a Yes. This function is always smaller, Czar, into your from one to infinity. One over x So e minus its power is it for this function? Denominator won over acts to ace power plus acts to humanise, eh is greeted on axe to be minus his power. These axes, critters and zero. Now look at this improper anti girl Face B is created on a past one. It's a B minus one. It's three to one. So this improper anti grow is commitment. And by the comparison, cereal thiss improper Anti Girl is also important is an integral from zero to infinity of the function axed weigh over one plus x two b's power is also converted.
Topics
Integration Techniques
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