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Chapter 6: Problem 40

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. $$ \int_{1}^{\infty} \frac{1}{x \ln x} d x $$

Short Answer

Expert verified
The integral diverges because the limiting value does not exist.

Step by step solution

01

Setting up the limit

Because the integral is improper (the upper limit is infinity), we change this to a limit problem. Let \( b\) be a number greater than 1. We will evaluate the limit as \( b\) approaches infinity for this integral: \( \lim_{b\rightarrow \infty}\int_{1}^{b} \frac{1}{x \ln x} dx \)
02

Integration by substitution

We can simplify the integral by using substitution. Let \( u = \ln x \). Then, \( du = \frac{1}{x} dx \). The integral is now: \( \lim_{b\rightarrow \infty}\int_{1}^{b} \frac{du}{u} \)
03

Evaluate the Integral

The integral \(\int \frac{du}{u}\) equals \(\ln |u|\). Therefore, \(\lim_{b\rightarrow \infty}\int_{1}^{b} \frac{du}{u} = \lim_{b\rightarrow \infty} \ln |u| = \lim_{b\rightarrow \infty} \ln |\ln x|\). We then evaluate this from 1 to \(b\), which gives \( \lim_{b\rightarrow \infty} (\ln |\ln b| - \ln |\ln 1|) \)
04

Evaluate the limit

\(\ln 1 = 0\), so that leaves us with \( \lim_{b\rightarrow \infty} \ln (\ln b) \). As b approaches infinity, \(\ln b\) also approaches infinity and therefore \(\ln (\ln b)\) approaches infinity. Therefore, the limit does not exist.

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Most popular questions from this chapter

In Exercises 65 and 66, apply the Extended Mean Value Theorem to the functions \(f\) and \(g\) on the given interval. Find all values \(c\) in the interval \((a, b)\) such that \(\frac{f^{\prime}(c)}{g^{\prime}(c)}=\frac{f(b)-f(a)}{g(b)-g(a)}\) \(\begin{array}{l} \underline{\text { Functions }} \\ f(x)=\ln x, \quad g(x)=x^{3} \end{array} \quad \frac{\text { Interval }}{\left[1,4\right]}\)

Show that the indeterminate forms \(0^{\circ}\) \(\infty^{0},\) and \(1^{\infty}\) do not always have a value of 1 by evaluating each limit. (a) \(\lim _{x \rightarrow 0^{+}} x^{\ln 2 /(1+\ln x)}\) (b) \(\lim _{x \rightarrow \infty} x^{\ln 2 /(1+\ln x)}\) (c) \(\lim _{x \rightarrow 0}(x+1)^{(\ln 2) / x}\)

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. $$ f(t)=e^{a t} $$

Consider the region satisfying the inequalities. (a) Find the area of the region. (b) Find the volume of the solid generated by revolving the region about the \(x\) -axis. (c) Find the volume of the solid generated by revolving the region about the \(y\) -axis. $$ y \leq e^{-x}, y \geq 0, x \geq 0 $$

Prove the following generalization of the Mean Value Theorem. If \(f\) is twice differentiable on the closed interval \([a, b],\) then \(f(b)-f(a)=f^{\prime}(a)(b-a)-\int_{a}^{b} f^{\prime \prime}(t)(t-b) d t\).
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