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

Calculate The Area Under The Graph Of Fx 1xlnx From X E To X Ee class=

Sagot :

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

How is the area under a graph found?

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

[tex]A = \int_{a}^{b} f(x) dx[/tex]

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

[tex]f(x) = \frac{1}{x\ln{x}}[/tex]

[tex]a = e, b = e^e[/tex]

Hence:

[tex]A = \int_{e}^{e^e} \frac{1}{x\ln{x}} dx[/tex]

Using substitution:

[tex]u = \ln{x}[/tex]

[tex]du = \frac{1}{x} dx[/tex]

[tex]dx = x du[/tex]

Hence:

[tex]\int \frac{1}{x\ln{x}} dx = \int \frac{1}{u} du = \ln{u} = \ln{\ln{x}}[/tex]

Applying the Fundamental Theorem of Calculus:

[tex]A = \ln{\ln{e^e}} - \ln{\ln{e}} = \ln{e\ln{e}} = \ln{\ln{e}} = \ln{e} - \ln{1} = 1[/tex]

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 https://brainly.com/question/20733870

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