Leeward 206 {8.13} Integral at infinity

In summary, the conversation is about solving the integral at infinity using the substitution method. The limits need to be changed accordingly and the substitution needs to be applied. The improper integral requires a finite terminal, which is why the variable "t" is introduced. This allows for the evaluation of the integral as the terminal approaches infinity.
  • #1
karush
Gold Member
MHB
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$\tiny\text{Leeward 206 {8.13} Integral at infinity}$
$$I=\int_{0} ^{\infty} e^{-ax} \,dx \ a>0 =
\\
\begin{align}\displaystyle
u& = -ax &
du&=-a \ d{x}
\end{align} \\
\text{then} \\
I=-\frac{1}{a}\int_{0} ^{\infty} e^{x} \,dx
=-\dfrac{\mathrm{e}^{-ax}}{a}+C \\
\text{hopefully, wasn't sure about the + C}$$
$\tiny\text{ Surf the Nations math study group}$
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  • #2
When you make the substitution you need to change your limits in accordance with the substitution. Let's begin with:

\(\displaystyle I=\lim_{t\to\infty}\left(\int_0^t e^{-ax}\,dx\right)\) where $0<a$

Now, if we use the substitution:

\(\displaystyle u=-ax\,\therefore\,du=-a\,dx\implies dx=-\frac{1}{a}\,du\)

and use the rule:

\(\displaystyle \int_a^b f(x)\,dx=-\int_b^a f(x)\,dx\)

We obtain:

\(\displaystyle I=\frac{1}{a}\lim_{t\to\infty}\left(\int_{-t}^0 e^{u}\,du\right)=\frac{1}{a}\lim_{t\to\infty}\left(\left[e^u\right]_{-t}^0\right)=\frac{1}{a}\lim_{t\to\infty}\left(1-e^{-t}\right)=\frac{1}{a}\)
 
  • #3
$\text{Why did you introduce t ?}$
 
  • #4
karush said:
$\text{Why did you introduce t ?}$

That's how I was taught to handle improper integrals. :)
 
  • #5
karush said:
$\text{Why did you introduce t ?}$

The reason Mark was taught that is how you handle improper integrals is because a definite integral requires FINITE values for the terminals of the integral. So by applying a finite terminal (in this case, "t") you can then evaluate the improper integral by seeing what happens as that finite value is increased without bound, in other words, what is the limiting value as $\displaystyle \begin{align*} t \to \infty \end{align*}$...
 

FAQ: Leeward 206 {8.13} Integral at infinity

1. What is the concept of "Leeward 206 {8.13} Integral at infinity"?

The concept of "Leeward 206 {8.13} Integral at infinity" refers to a mathematical concept in calculus where the limit of a function is taken as the input variable approaches infinity. This concept is used to determine the behavior of a function as the input variable becomes larger and larger.

2. How is the "Leeward 206 {8.13} Integral at infinity" calculated?

The "Leeward 206 {8.13} Integral at infinity" is calculated by taking the limit of a function as the input variable approaches infinity. This can be done using mathematical techniques such as L'Hopital's rule or by using specific integration techniques for functions that are known to have infinite limits.

3. What is the significance of the "Leeward 206 {8.13} Integral at infinity" in mathematics?

The "Leeward 206 {8.13} Integral at infinity" is significant in mathematics as it allows us to determine the long-term behavior of a function. It helps us understand how a function behaves as the input variable becomes very large, which can be useful in many real-world applications such as modeling population growth or predicting the behavior of complex systems.

4. Can the "Leeward 206 {8.13} Integral at infinity" be calculated for all functions?

No, the "Leeward 206 {8.13} Integral at infinity" can only be calculated for certain types of functions. It cannot be calculated for functions that do not have infinite limits, such as polynomial functions or functions with finite limits as the input variable approaches infinity.

5. What are some common applications of the "Leeward 206 {8.13} Integral at infinity"?

The "Leeward 206 {8.13} Integral at infinity" has many applications in various fields of mathematics, science, and engineering. Some common applications include determining the long-term behavior of population growth models, analyzing the stability of complex systems, and understanding the convergence of series in calculus.

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