Help with Iterated Integrals: Solving ln(2+y)-ln(1+y) dy [0,1]

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In summary, an iterated integral is a type of integral that involves multiple variables and is used to find the volume under a surface or the area between curves. It is typically expressed as the integral of an integral, with one integral being nested inside the other. To solve an iterated integral, we can use the properties of logarithms and the substitution method. The upper limit of the integral is determined by the given upper bound in the interval. It is possible to use a calculator to solve iterated integrals, but it is important to understand the concepts and steps involved. The natural logarithm is used in this integral as it is the inverse of the exponential function and simplifies the integral. It also has many applications in mathematics and science.
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izelkay
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Homework Statement


Here's a screenshot of the solved integral by wolfram: http://puu.sh/4CIzy.png

The Attempt at a Solution



I get stuck about halfway through working through this, with ∫ln(2+y)-ln(1+y) dy [0,1]
My problem is I'm not sure how to integrate ln(2+y) or ln(1+y), since integration by parts doesn't seem to work for either of them, and if I had integrated in terms of y first I'd be left with similar, hard to integrate natural logs for x. Can someone walk me through how to integrate these?
 
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Never mind, figured it out.
 

1. What is an iterated integral?

An iterated integral is a type of integral that involves multiple variables and is used to find the volume under a surface or the area between curves. It is typically expressed as the integral of an integral, with one integral being nested inside the other.

2. How do you solve ln(2+y)-ln(1+y) dy [0,1]?

To solve this iterated integral, we first need to convert it into a single integral by using the properties of logarithms. This will give us ln((2+y)/(1+y)) dy. Then we can use the substitution method, letting u = (2+y)/(1+y) and du = dy/(1+y). After substituting and simplifying, we can integrate with respect to u and then substitute back in for y to get our final answer.

3. Why is the upper limit of the integral 1?

The upper limit of the integral is 1 because it is the given upper bound in the interval [0,1]. This means that we are finding the area under the curve from x=0 to x=1.

4. Can you use a calculator to solve iterated integrals?

Yes, you can use a calculator to solve iterated integrals. However, it is important to understand the concepts and steps involved in solving them by hand before relying on a calculator.

5. What is the significance of the natural logarithm in this integral?

The natural logarithm, ln, is used in this integral because it is the inverse of the exponential function e^x. In this case, we are using it to simplify the integral and make it easier to solve. The natural logarithm also has many applications in mathematical and scientific fields.

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