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Mekhanik [1.2K]
2 years ago
15

Calculate the area under the graph of f(x) = 1/xlnx from x = e to x = e^e

Mathematics
1 answer:
Gnom [1K]2 years ago
7 0

Using integration, it is found that the area under the graph of f(x) in the desired interval is of 1 unit squared.

<h3>How is the area under a graph found?</h3>

The area under the graph of a curve f(x) between x = a and x = b is given by the following integral:

A = \int_{a}^{b} f(x) dx

In this problem, the function and the limits of integration are given by:

f(x) = \frac{1}{x\ln{x}}

a = e, b = e^e

Hence:

A = \int_{e}^{e^e} \frac{1}{x\ln{x}} dx

Using substitution:

u = \ln{x}

du = \frac{1}{x} dx

dx = x du

Hence:

\int \frac{1}{x\ln{x}} dx = \int \frac{1}{u} du = \ln{u} = \ln{\ln{x}}

Applying the Fundamental Theorem of Calculus:

A = \ln{\ln{e^e}} - \ln{\ln{e}} = \ln{e\ln{e}} = \ln{\ln{e}} = \ln{e} - \ln{1} = 1

Hence, the area under the graph of f(x) in the desired interval is of 1 unit squared.

To learn more about integration, you can take a look at brainly.com/question/20733870

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