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Solve the given problems by integration.Show that $\int_{0}^{3} \frac{d x}{2 x+2}=\ln 2$.

Calculus 1 / AB

Chapter 28

Methods of Integration

Section 2

The Basic Logarithmic Form

Integrals

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Evaluate the integral.…

we want to evaluate the definite role flog Bates to of X Plus two, divided by experts to from 0 to 2. Now notice that we have a log and then we have that long being divided by the thing on the inside of the lock. So any time we see something like that that it's a good idea to maybe try a U substitution of the log and then seeing if we can get some things to come slow. So we let you equal to log base to express to. And then we take the derivative of this with respect to X to get the differential d'you is equal to. So the derivative of the log will be one over x times the natural log of the base. So normally I'd have an X here. But since on the inside of that is exposed to, we're gonna have that there and then the natural log of the base, too. And then remember. Also, since this is a function in a function, we would need to take the derivative of X plus two as well. But in that case, review of exes Wonder driven to zero, so not zero, but one. And then lastly, we would just throw on r d X difference. So if we multiply the natural log of two over, we get natural longer too. Do you? Is equal to DX over X plus two. And it just so happens way Have those two there so we can replace the DX over, exposed to by the natural log of two, do you? And then our log is just going to be you. And now we can go ahead and change our Alan's of integration. And we will do that by using what are U substitution? Wallace. So we need to use this here. So when X is equal to zero, well, we plug and zero. So we get log base to up to, and that just so happens to be one. So our new lower bound will be one. And then when X is equal to two, we get you is even too log base to a four, and that's two. It's our new over bound. Or I guess our upper body states same being too. All right now, we can go ahead and background that natural longer, too, and when we do that, we'd be left with in a roll from one the two of you, do you? And to integrate you, we would use power rule. So natural log of two. This would all be you to the second power. And we divide all this but you And then we evaluate from 1 to 2 so we can factor out that natural August too. And they go ahead and plug in to infer you squared, which gives us four and then subtract off when we plug in one into you squared, which is gives us one. So we get four miles on its three and then we can write that as three natural log of two over too. So this is what we get when we evaluate this definite.

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