Integral from e to infinity

From here, it is clear that the integral converges, since the integrand approaches 0 as u approaches infinity. The antiderivative of \frac{1}{u^2} is -\frac{1}{u}, so the integral evaluates to -\frac{1}{2} \left.\frac{1}{u}\right|_1^\infty = -\frac{1}{2} \left(0 - \frac{1}{1}\right) = \frac{1}{2}. Therefore, the overall value of the integral is \frac{67}{2}, as you stated. In summary, the integral from e to infinity of 67 / (x(ln(x))^3 is conver
  • #1
tnutty
326
1

Homework Statement



integral [from e to infinity of ] 67 / (x(ln(x))^3)

read as the integral from e to infinity of

67 divided by x times cubed lnx.


Homework Equations





The Attempt at a Solution





i know it converges,

but i got the value

67/2
 
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  • #2


tnutty said:

Homework Statement



integral [from e to infinity of ] 67 / (x(ln(x))^3)

read as the integral from e to infinity of

67 divided by x times cubed lnx.

The Attempt at a Solution



i know it converges,

but i got the value

67/2
And I would agree with you...

Using the substitution [tex]u=\ln{x}[/tex], the integral

[tex]\int_e^\infty \frac{1}{x(\ln{x})^3} \, dx[/tex]

becomes

[tex]-\frac{1}{2} \int_1^\infty \frac{1}{u^2} \, du.[/tex]
 

What is the meaning of "Integral from e to infinity"?

The integral from e to infinity is a mathematical concept that represents the area under a curve from the value of e to infinity. It is typically written as ∫e^x dx and is used in calculus to solve problems related to rates of change and accumulation.

How is the integral from e to infinity calculated?

The integral from e to infinity is usually calculated using integration by parts or substitution, as it involves solving an infinite limit. It can also be approximated using numerical methods, such as the trapezoidal rule or Simpson's rule.

What is the significance of using e in the integral from e to infinity?

The number e (approximately 2.718) is a special mathematical constant that appears frequently in natural and exponential growth. Using e in the integral from e to infinity allows us to solve problems related to exponential functions and continuously growing quantities.

Why is the integral from e to infinity important in mathematics?

The integral from e to infinity is an essential tool in calculus and mathematical analysis. It is used to solve various problems related to rates of change, growth, and accumulation, making it a fundamental concept in many areas of mathematics and science.

Are there any real-world applications of the integral from e to infinity?

Yes, the integral from e to infinity has a wide range of real-world applications, such as in physics, engineering, and economics. For example, it can be used to calculate the total amount of heat generated over time, the population growth of a species, or the accumulated interest on a continuously compounding investment.

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