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JH

# Evaluate the integral.$\displaystyle \int_1^4 \sqrt{y} \ln y\ dy$

## $\frac{16}{3} \ln 4-\frac{28}{9}$

#### Topics

Integration Techniques

### Discussion

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##### Kristen K.

University of Michigan - Ann Arbor

##### Samuel H.

University of Nottingham

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### Video Transcript

Let's use integration by parts for this in a barrel. Let's take you to be natural, Aga boy. Then do you one of her lie D y? And this leaves us with Devi equal squared or why d y and taking the integral here will give us two thirds. Why the three house? So using our formula here This me right on the side, juvie minus and a girl video. So you times V in the wrong order. But there they are And then our end points once before minus in a girl VDO So let me pull out the two thirds And then after cancelling the wise, we have one half. So let's just go ahead and evaluate this term first. So, Ellen, for times two or three and then for the three halfs and then minus zero after you plug in one minus and we've already evaluated this inaugural. So here we should get to over three coming from over here and then another two of three coming from here. Why three house once for so here we have squared off form, which is too then to the third power, which is eight and then times two, so That's sixteen for the first term. And then here. Let's go ahead and simplify this. So we get full over nine and then we got a plug in foreign one. So here, sixteen over three. And then we've already evaluated this to be eight. So take off one and then multiply it by four. So we have twenty eight up there, all divided by nine. How? And that's your final answer.

JH

#### Topics

Integration Techniques

##### Kristen K.

University of Michigan - Ann Arbor

##### Samuel H.

University of Nottingham

Lectures

Join Bootcamp