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Determine whether each integral is convergent or divergent. Evaluate those that are convergent.$$\int_{0}^{\infty} \frac{1}{\sqrt[4]{1+x}} d x$$

integral is divergent

Calculus 2 / BC

Chapter 6

TECHNIQUES OF INTEGRATION

Section 6

Improper Integrals

Integration Techniques

Campbell University

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University of Michigan - Ann Arbor

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Lectures

01:11

In mathematics, integratio…

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Determine whether each int…

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in this problem were given Thean proper integral one over the quantity one plus x two, the 1/4 from zero to infinity and were asked to determine whether this is a convergent or divergent Integral. So to do this, I'm going to take that infinity bound. And instead I'm going to write it as the limit as T approaches infinity of t 20 of 1/1 plus x two, the 1/4 DX and now we can solve this. So I'm gonna go ahead and begin by just worrying about our indefinite, integral installing for that and then we can plug in the bounds. So if we solve this, we get 4/3 times the quantity one plus x to the 3/4 plus C. If we go ahead and plug in our definite bounds into this, we get 4/3 times a quantity one plus x to the 3/4 from zero to t. If you go ahead and plug in these bounds, I'm going to factor out that 4/3 and put it in the front and we get the quantity one plus t 3/4 minus one. So now all we have to do is take the limit of this as T approaches infinity of 4/3 times one plus t 3/4 minus one And if we do this will notice that we can also right this as 4/3 times infinity to the 3/4 minus one. And this is just going to leave us with infinity. And because of that, we know that this integral has to be divergent.

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