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Problem 37 Hard Difficulty

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

$ \displaystyle \int_0^1 r \ln r\ dr $


0, Convergent


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Video Transcript

the problem is determine whether each integral is coordinated. Word of urgent. You violated those that word. It thiss improper, integral that nation. This is equal to the limit. A ghost, tio zero from right hand side into go from a one r Ellen are the look at it. Is this definite? Integral to Stephanie being the girl Eco's indeed girl from a one, you know. And, uh yeah. One, huh? Ask for here. We use, um my third of integration. My parts staffing the integral is he called Tio. You are one our square. Some es juan minus into girl remained one one half square. One over r yeah, is because two flying one on a to this function. This is zero minus, eh? What a squire, my ass. This's while half off times are so on a derivative. Off this function is one force, uh, square from a one. This recall to make our end, eh, Holmes? One Hey, square minus one. Force minus one. Force a square. Hear what a goes to zero. A square goes to zero this far because two zero on ourselves This part ln aid hams. A square also goes to zero for this card, you can use NOPD toe computed. You go to a close to zero, you know, And eh Over one terms, one over a square. Then we use a low paid US rule. This is Nico to limit one over, eh? Over. This's a native one over to kill. It goes to zero. This is equal to zero. No. So the result is zero minus one Force. This is making you force. This improper into girl is confident on the value is ninety one over four.