Chapter 11: Q10.6P (page 552)

The logarithmic integralis $l i (x) = \int_{0}^{x} \frac{dt}{In t}$ . Express as exponential integrals
$l i (x)$
$l i (e^{x})$
$l i (x) = \int_{0}^{x} \frac{dt}{In (\frac{1}{t})}$

Short Answer

Expert verified

The following required expressions are shown:

$l i (x) = E_{i} (I n x)$
$l i (e^{x}) = E_{i} (x)$
$= - E_{i} (I n x)$

Step by step solution

Given information

The definition of a logarithmic integral is given.

$l i (x) = \int_{0}^{x} \frac{dt}{In t}$

Begin with making a substitution.

(a)

Make the following substitution.

$\begin{array}{rcl} t & = & e^{u} \\ d t & = & e^{u} d u \end{array}$

Apply the substitution in the given integrand.

$\begin{array}{rcl} l i (x) & = & \int_{- \infty}^{I n (x)} \frac{e^{u}}{u} d u \\ = & E_{i} (I n x) \end{array}$

Substitute the values in the domain.

(b)

Substitute e^x in the domain of li(x).

$\begin{array}{rcl} l i (e^{x}) & = & \int_{- \infty}^{x} \frac{e^{u}}{u} d u \\ = & E_{i} (x) \end{array}$

Repeat the process.

(c)

Use the following substitution.

$\begin{array}{rcl} t & = & e^{u} \\ d t & = & e^{u} d u \end{array}$

Apply the substitution in the given integrand.

$\begin{array}{rcl} \int_{- \infty}^{I n x} \frac{e^{u}}{- u} d u & = & \int_{I n (x)}^{- \infty} \frac{e^{u}}{u} d u \\ = & - E_{i} (I n x) \end{array}$

Thus, the following expressions are shown:

$(a) l i (x) = E_{i} (I n x) (b) l i (e^{x}) = E_{i} (x) (c) = - E_{i} (I n x)$

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The logarithmic integralis $l i (x) = \int_{0}^{x} \frac{dt}{In t}$ . Express as exponential integrals
$l i (x)$
$l i (e^{x})$
$l i (x) = \int_{0}^{x} \frac{dt}{In (\frac{1}{t})}$

Short Answer

Step by step solution

Given information

Begin with making a substitution.

Substitute the values in the domain.

Repeat the process.

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The logarithmic integralis li(x)=∫0xdtInt. Express as exponential integralsli(x)li(ex)li(x)=∫0xdtIn(1t)

Short Answer

Step by step solution

Given information

Begin with making a substitution.

Substitute the values in the domain.

Repeat the process.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Famous Physicists

Geometrical and Physical Optics

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Classical Mechanics

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Study anywhere. Anytime. Across all devices.

The logarithmic integralis $l i (x) = \int_{0}^{x} \frac{dt}{In t}$ . Express as exponential integrals
$l i (x)$
$l i (e^{x})$
$l i (x) = \int_{0}^{x} \frac{dt}{In (\frac{1}{t})}$