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Use the Substitution Formula in Theorem 7 to evaluate the integrals.$$\int_{1}^{2} \frac{2 \ln x}{x} d x$$

$ln^2(2)$

Calculus 1 / AB

Chapter 5

Integrals

Section 6

Definite Integral Substitutions and the Area Between Curves

Integration

Campbell University

University of Michigan - Ann Arbor

University of Nottingham

Lectures

03:09

In mathematics, precalculu…

31:55

In mathematics, a function…

02:06

Use the Substitution Formu…

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we want to use a substitution formula in there. Um, seven to help us evaluate this integral here. Now, one thing you might notice is that we have the natural X. We know the derivative of the natural log of X is one over X. So maybe if we let you be natural log of X, our differential should give us one of her acts DX. And that's exactly what we have right there. So we could just go ahead and replace that with to you d'you and then our new bounds will be Well, let's go ahead and plug in one and two. So when X isn't one that's gonna give us you is you to the natural log of one which is zero. And then when X is equal to two that would give us you is equal to the natural log of two and can't really simplify that much more. So just go ahead and leave it like that. Now we know how to integrate just you. That's just power rule for integration. So to be two times you to the second, and then we divide by its new power, which is to and evaluate from zero to natural law goes to. So the twos here will cancel out with each other, and then we just need to plug in the natural order two and zero. So plug it in natural log to that's gonna give us the natural log of two squared and then minus zero. So we're just left with the natural log of two squared.

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