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Chapter 11: Problem 13

Express each of the following integrals as a \(\Gamma\) function. By computer, evaluate numerically both the \(\Gamma\) function and the original integral. \(\int_{0}^{1} x^{2}\left(\ln \frac{1}{x}\right)^{3} d x\) Hint: Put \(x=e^{-u}\)

Short Answer

Expert verified
\(\frac{2}{27}\)

Step by step solution

01

- Substitution

Use the substitution given in the hint: Let \(x = e^{-u}\). Then, \(dx = -e^{-u} du\).
02

- Change of limits

Determine the new limits of integration for the variable \(u\). When \(x = 0\), \(u = \infty\). When \(x = 1\), \(u = 0\). Thus, the integral is now \( \int_\infty^0 x^{2} \left( \ln \frac{1}{x} \right)^{3} (-e^{-u} du)\).
03

- Simplify the integrand

Substitute \(x = e^{-u}\) into the integrand: \(x^2 = (e^{-u})^2 = e^{-2u}\) and \(\ln \frac{1}{x} = \ln e^u = u\). Now the integral becomes \( \int_ \infty ^0 e^{-2u} u^3 (-e^{-u}) du\).
04

- Integrate

Simplify the integral: \(\int_{\infty}^{0} e^{-3u} u^3 (-du) = \int_{0}^{\infty} e^{-3u} u^3 du\). This integral is in the form of a Gamma function: \(\Gamma(n) = \int_{0}^{\infty} t^{n-1} e^{-t} dt\). Rewrite the integral as \(\int_{0}^{\infty} u^{3} e^{-3u} du = \frac{1}{3^4} \int_{0}^{\infty} (3u)^3 e^{-(3u)} d(3u)\).
05

- Express in Gamma function

Recognize that the expression is now: \(\frac{1}{3^4} \int_{0}^{\infty} t^3 e^{-t} dt\), where \(t = 3u\). According to the definition of the Gamma function, this is \(\frac{1}{3^4} \Gamma(4)\) as \(\Gamma(n) = (n-1)!\).
06

- Evaluate the Gamma function

Calculate \(\Gamma(4)\): \(\Gamma(4) = 3! = 6\). Thus, the integral becomes \(\frac{1}{3^4} * 6 = \frac{6}{81} = \frac{2}{27}\). Verify this using a computational tool to ensure accuracy.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Gamma function
The Gamma function, denoted as \( \Gamma(n) \), is a continuous extension of the factorial function to the real and complex plane. Specifically, for any positive integer \( n \), \( \Gamma(n) = (n-1)! \). It is defined via an improper integral: \

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Most popular questions from this chapter

Computer plot graphs of \(\operatorname{sn} u,\) cn \(u,\) and \(\mathrm{dn} u,\) for several values of \(k,\) say, for example, \(k=1 / 4,1 / 2,3 / 4,0.9,0.99 .\) Also plot \(3 \mathrm{D}\) graphs of \(\mathrm{sn}, \mathrm{cn},\) and \(\mathrm{dn}\) as functions of \(u\) and \(k\).

Identify each of the following integrals or expressions as one of the functions of this chapter. Check your work by evaluating both your answer and the original problem by computer. Be sure you understand your computer program's notation. $$\int_{0}^{7 \pi / 8} \sqrt{4-\sin ^{2} x} d x$$

Identify each of the following integrals or expressions as one of the functions of this chapter. Check your work by evaluating both your answer and the original problem by computer. Be sure you understand your computer program's notation. $$\int_{-\pi / 4}^{3 \pi / 4} \frac{d \phi}{\sqrt{1+\cos ^{2} \phi}}$$

Identify each of the following integrals or expressions as one of the functions of this chapter. Check your work by evaluating both your answer and the original problem by computer. Be sure you understand your computer program's notation. $$\int_{0}^{1} e^{-x^{2}} d x$$

Show that for \(k=0\): $$u=F(\phi, 0)=\phi, \quad \operatorname{sn} u=\sin u, \quad \operatorname{cn} u=\cos u, \quad \operatorname{dn} u=1$$ and for \(k=1\): $$\begin{array}{c} u=F(\phi, 1)=\ln (\sec \phi+\tan \phi) \quad \text { or } \quad \phi=\operatorname{gd} u \quad \text { (Problem } 19) \\ \quad \operatorname{sn} u=\tanh u, \quad \operatorname{cn} u=\operatorname{dn} u=\operatorname{sech} u \end{array}$$
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