/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 22 Determine whether the improper i... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 22

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges. $$ \int_{1}^{\infty} \frac{\ln x}{x} d x $$

Short Answer

Expert verified
The improper integral \( \int_{1}^{\infty} \frac{\ln x}{x} \, dx \) converges and its value is 0.

Step by step solution

01

Identify the Integral

Recognize that the given integral \( \int_{1}^{\infty} \frac{\ln x}{x} d x \) is an improper integral of the first kind because the interval of integration is infinite.
02

Introduce limit notation

Rewrite the integral using limit notation, replacing the upper limit of integration, infinity, with a variable, \( t \), and taking the limit as \( t \) approaches infinity. This transforms the original integral into a limit of proper integrals: \( \lim_{t \to \infty} \int_{1}^{t} \frac{\ln x}{x} dx \)
03

Use integration by parts

Now, apply the method of integration by parts to evaluate the integral. This method is suggested by the form of the integrand, which is the product of \( \ln x \) and \( \frac{1}{x} \). Using the formula, the integral becomes: \( \lim_{t \to \infty} [-\ln t + \int_{1}^{t} \frac{1}{x} dx] \)
04

Simplify further

Perform further simplification. The natural logarithm integral can then be processed as follows: \( \lim_{t \to \infty} [-\ln t + \ln|x|]_{1}^{t} \)
05

Applying limits

Now the integral should be solved for the desired area. Substitute the upper and lower limits of integration锛坱 and 1锛塮or \( x \) respectively: \( \lim_{t \to \infty} [(\ln t - \ln t) - (\ln 1 - \ln 1)] = \lim_{t \to \infty} [0]\)
06

Evaluating limit

Evaluate this limit as \( t \) approaches infinity, which results in 0.

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Most popular questions from this chapter

Laplace Transforms Let \(f(t)\) be a function defined for all positive values of \(t\). The Laplace Transform of \(f(t)\) is defined by \(F(s)=\int_{0}^{\infty} e^{-s t} f(t) d t\) if the improper integral exists. Laplace Transforms are used to solve differential equations. Find the Laplace Transform of the function. $$ f(t)=t $$

Graphical Analysis In Exercises \(\mathbf{6 1}\) and 62, graph \(f(x) / g(x)\) and \(f^{\prime}(x) / g^{\prime}(x)\) near \(x=0 .\) What do you notice about these ratios as \(x \rightarrow 0\) ? How does this illustrate L'H么pital's Rule? \(f(x)=\sin 3 x, \quad g(x)=\sin 4 x\)

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Use a graphing utility to graph \(f(x)=\frac{x^{k}-1}{k}\) for \(k=1,0.1\), and 0.01 . Then evaluate the limit \(\lim _{k \rightarrow 0^{+}} \frac{x^{k}-1}{k}\).
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