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Solve the given problems by integration.Find the first-quadrant area bounded by $y=\frac{\ln (4 x+1)}{4 x+1}$ and $x=5$.

Calculus 1 / AB

Chapter 28

Methods of Integration

Section 1

The General Power Formula

Integrals

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given these three functions, the area bounded by all three. And the first quadrant is this shaded black area here, and we'll go ahead and set up multiple into girls. First off this intersection, point has an X value of 0.5 and this intersection point as, ah x value of to. And of course, the two of those functions will be origin there. And so the first integral that will set up for the area under the current is from 0 to 0.5 of the top function, which is four x minus bottom function, which is X over four DX, and that will get us the area on the left side here from zero to 0.5. So all that shaded, uh, darker black area And then from here, we don't want to add the area under the curve from 0.5 to to and the top function keer now is one over X. Subtract the bottom function, which is X over four, and these will be the wits de exes times, um, everything in parentheses there is the height of all of our rectangles. So there's all that. That'll add the exact amount of area in this second part and then, in the end, taking on 10 derivatives here, the they're all just polynomial so that those steps are fairly straightforward. There we'll end up with one point three eight six to nine. The reason why it's a little funky is because when we anti drive one over X, we can end up a force with natural log and so that it's gonna have some some, uh, irrational numbers. And so this is approximately our final answer for the total area on between all three of those curves.

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