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Magazine Chapter 5: Problem 22
Determine whether the improper integral is convergent or divergent, and calculate its value if it is convergent. \(\int_{0}^{\infty} \frac{e^{t}}{\left(5+e^{t}\right)^{2}} d t\)
Short Answer
Expert verified
The integral is convergent and its value is \(\frac{1}{5}\).
Step by step solution
01
Identify the Type of Improper Integral
The given integral is \(\int_{0}^{\infty} \frac{e^{t}}{(5+e^{t})^{2}} dt\). Since the upper limit is infinity, it's an improper integral of the second kind.
02
Apply the Limit Definition for Improper Integrals
Rewrite the integral in terms of a limit to handle the infinity: \[\int_{0}^{\infty} \frac{e^{t}}{(5+e^{t})^{2}} dt = \lim_{b \to \infty} \int_{0}^{b} \frac{e^{t}}{(5+e^{t})^{2}} dt.\]
03
Simplify the Integrand
Let \(u = 5 + e^t\). Then \(du = e^t dt\). When \(t=0\), \(u=5\). When \(t \to \infty\), \(u \to \infty\). The integral becomes: \[\lim_{b \to \infty} \int_{0}^{b} \frac{e^{t}}{(5+e^{t})^{2}} dt = \lim_{c \to \infty} \int_{5}^{c} \frac{1}{u^2} du\]
04
Integrate
Find the antiderivative of \(\frac{1}{u^2}\). The antiderivative is \(-\frac{1}{u}\). So, \[\int_{5}^{c} \frac{1}{u^2} du = \left. -\frac{1}{u} \right|_{5}^{c} = \left(-\frac{1}{c} + \frac{1}{5}\right)\]
05
Evaluate the Limit
Evaluate the limit as \(c \to \infty\): \[\lim_{c \to \infty} \left(-\frac{1}{c} + \frac{1}{5}\right) = 0 + \frac{1}{5} = \frac{1}{5}.\]
06
Conclude the Convergence
Since the limit exists and is finite, the improper integral is convergent. The value of the integral is \(\frac{1}{5}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Improper Integrals
An **improper integral** is an integral where either the interval of integration is infinite, or the integrand becomes infinite within the interval of integration. In our given example, \(\begin{equation*}\begin{split}\begin{aligned}\int_{0}^{\infty} \frac{e^{t}}{(5+e^{t})^{2}} dt, \text{\end{aligned\)\text{\(\) e are the properties that make the integral improper.Improper integrals often require special techniques to evaluate or determine their convergence. One common approach is transforming the integral using the limit definition.\
- This specific integral is improper because the upper limit of integration is infinity.
- We apply limits to handle the infinite boundary.
Limit Definition
The **limit definition** is essential when dealing with improper integrals. Instead of calculating the integral directly over an infinite interval, we approach it by breaking it down into a limit process. For our exercise, the limit definition transforms the problem as follows:\[\int_{0}^{\infty} \frac{e^{t}}{(5+e^{t})^{2}} dt = \lim_{b \to \infty} \int_{0}^{b} \frac{e^{t}}{(5+e^{t})^{2}} dt.\]
- Using the limit definition, we replace the infinity symbol with a variable, say \(b\).
- Then, we take the limit of the integral as \(b\) approaches infinity.
Substitution Method
One of the powerful tools for simplifying integrals is the **substitution method**. For our example, we use the substitution \(u = 5 + e^t\). The steps are:
- Rewrite the integrand in terms of \(u\).
- Calculate the derivative: if \(u = 5 + e^t\), then \(du = e^t dt\).
- Adjust the limits of integration accordingly: when \(t = 0\), \(u = 5\), and when \(t \to \infty\), \(u \to \infty\).
Antiderivative
The **antiderivative** is the reverse process of differentiation and is crucial in solving integrals. For the integrand \(\frac{1}{u^2}\),
- The antiderivative is found to be \(-\frac{1}{u}\).
Convergence and Divergence
Finally, we determine whether the improper integral converges or diverges. For convergence,
- The limit must exist and be finite.
- If so, the integral is said to be convergent. If not, it is divergent.
