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Problem 27 Medium Difficulty

Evaluate the indefinite integral as a power series. What is the radius of convergence?
$ \int x^2 \ln (1 + x) dx $

Answer

$$C+\sum_{n=1}^{\infty} \frac{(-1)^{n} x^{n+3}}{n(n+3)}, \mathrm{R}=1$$

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Video Transcript

Okay, so you batter the infinite, Integral as a power. Siri's and fun is realism. Convergence. All right, so we can first expend lawn if one plus x. So this is just X minus thanks. Square over two plus x cuba or three minus. That's the fourth over four. And so alone. So the ex assistant X square tons. That's over N to the power. In terms of one on one, it's one trip out on minus one, and it is from one to infinity and this whole thing, the eggs and we change the world and the sub so it becomes and from zero to and from one divinity and x to the power and plus two or in terms that you want to go over. A nice one, the ex and yeah, the universe inside the summation. All right. And what is Thie? And they were off this parts we can pull out now if you want to power. And one is one that we're in sure is from one to even any. And we single you Ballard any roof next to help us to the X. This is just over in times. So thanks to you, power and industry over in three. Yes, so this is our final answer. Forward the power serious expand like you want times and that you want to power and mothers one times extra ball from this three over in times in three and fun. It's, um, readers of convergence. Okay, so where the where the way spend it so expanded from here. Yes, so we could remember the readers of Converters of London because actually, Ari Koh's won and it just applies for this new series. It's called Max. So the readers of Half of X is just Bari who's won.