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WZ
Numerade Educator

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Problem 17 Medium Difficulty

Determine whether each integral is convergent or divergent. Evaluate those that are convergent.

$ \displaystyle \int_1^\infty \frac{1}{x^2 + x}\ dx $

Answer

convergent to $\ln 2$

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Video Transcript

the problem is determine whether this integral is cordant or divergent the first by definition, of an improper integral. This is equal to the limit. He goes to infinity. That's integral from one to t bond over X square US X, the axe Mhm. And this is echo to limit. He goes to infinity Integral 12 t We can rewrite this function as one over acts times X plus one. Yeah. Now this is the echo to the limit He goes to infinity Integral one, the T Also, we can rewrite this function as one over X minus one over X as one the X Now we can write this. This is equal to the limit. He goes to infinity as anti derivative of one Iraq's Ln acts This anti derivative of one over X plus one is Ln X plus one from one to keep. This is echo is the limit He goes to infinity Ln X over X plus one and wanted to. This is a co two limit. He goes to infinity Alan she over walk one last T minus now and 1/2. When he goes to infinity, she over one plus t goes to one, so l m t over one plus t goes to zero. The answer is equal to negative Ln one half. And this is also the co two Ln two mhm. So this integral is concordant and this value is l N two.