143.
b. Find the average value of ln(x) over [1, e].

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Recall the formula for the average value of a function $f(x)$ over the interval $[a, b]$: \[\text{Average value} = \frac{1}{b - a} \int_a^b f(x) \, dx\]

Identify the function and interval: here, $f(x) = \ln(x)$, $a = 1$, and $b = e$.

Set up the integral for the average value: \[\frac{1}{e - 1} \int_1^e \ln(x) \, dx\]

To evaluate the integral $\int \ln(x) \, dx$, use integration by parts. Let $u = \ln(x)$ so that $du = \frac{1}{x} dx$, and let $dv = dx$ so that $v = x$.

Apply integration by parts formula: \[\int u \, dv = uv - \int v \, du\] which gives \[\int \ln(x) \, dx = x \ln(x) - \int x \cdot \frac{1}{x} \, dx = x \ln(x) - \int 1 \, dx = x \ln(x) - x + C\] Use this to evaluate the definite integral from 1 to $e$ and then substitute back into the average value formula.

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Concetti chiave

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Average Value of a Function

The average value of a continuous function f(x) over an interval [a, b] is given by (1/(b - a)) times the integral of f(x) from a to b. It represents the mean height of the function on that interval.

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A definite integral calculates the net area under the curve of a function between two points a and b. It is essential for finding accumulated quantities, such as total area or average values over intervals.

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Definition of the Definite Integral

Integration of the Natural Logarithm Function

Integrating ln(x) requires integration by parts, using u = ln(x) and dv = dx. The integral of ln(x) is x ln(x) - x + C, which is crucial for evaluating the definite integral of ln(x) over [1, e].

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