Problem 17
Question
Determine whether each integral is convergent. If the integral is convergent, compute its value. $$ \int_{0}^{\infty} \frac{1}{x+1} d x $$
Step-by-Step Solution
Verified Answer
The integral diverges as it approaches infinity.
1Step 1: Set Up the Integral
We are given the improper integral \( \int_{0}^{\infty} \frac{1}{x+1} \, dx \). Since it is an improper integral due to the upper limit being infinity, we will have to evaluate it as a limit: \( \lim_{b \to \infty} \int_{0}^{b} \frac{1}{x+1} \, dx \).
2Step 2: Find the Antiderivative
To find the antiderivative of \( \frac{1}{x+1} \), let \( u = x+1 \). Thus, \( du = dx \). The antiderivative of \( \frac{1}{u} \) is \( \ln|u| \). Substituting back, we have \( \ln|x+1| \).
3Step 3: Evaluate the Definite Integral from 0 to b
Now we compute the definite integral from 0 to \( b \): \( \int_{0}^{b} \frac{1}{x+1} \, dx = \bigg[ \ln|x+1| \bigg]_{0}^{b} = \ln(b+1) - \ln(1) = \ln(b+1) \).
4Step 4: Evaluate the Limit
Evaluate the limit \( \lim_{b \to \infty} \ln(b+1) \). As \( b \to \infty \), \( \ln(b+1) \to \infty \). Thus, the integral \( \int_{0}^{\infty} \frac{1}{x+1} \, dx \) diverges.
Key Concepts
Convergence of IntegralsLimit EvaluationAntiderivative Calculation
Convergence of Integrals
In calculus, the concept of convergence is essential when working with improper integrals. An improper integral can have limits that go to infinity or involve a function that is unbounded. To determine if such an integral is convergent, one needs to assess whether its limit exists and gives a finite value.
For instance, if we consider the integral \[ \int_{0}^{\infty} \frac{1}{x+1} \, dx \]we must analyze whether it approaches a certain number as the upper limit goes to infinity. This involves evaluating whether the improper behavior at infinity does not lead to an unmanageable or infinite result.
If the integral exhibits this kind of controlled behavior and approaches a finite limit, it is said to be convergent. Otherwise, it is divergent, meaning it does not settle to a single finite value as it stretches towards infinity. This basic distinction is crucial for interpreting the results and implications of evaluating improper integrals.
For instance, if we consider the integral \[ \int_{0}^{\infty} \frac{1}{x+1} \, dx \]we must analyze whether it approaches a certain number as the upper limit goes to infinity. This involves evaluating whether the improper behavior at infinity does not lead to an unmanageable or infinite result.
If the integral exhibits this kind of controlled behavior and approaches a finite limit, it is said to be convergent. Otherwise, it is divergent, meaning it does not settle to a single finite value as it stretches towards infinity. This basic distinction is crucial for interpreting the results and implications of evaluating improper integrals.
Limit Evaluation
A key step in handling improper integrals is the evaluation of limits. By transforming an integral with an infinite limit into a limit statement, we can tackle its infinity aspect more directly. For example, the improper integral \[ \int_{0}^{\infty} \frac{1}{x+1} \, dx \]can be rewritten using a limit: \[ \lim_{b \to \infty} \int_{0}^{b} \frac{1}{x+1} \, dx \].
This formulation lets us investigate what happens to the integral as the boundary extends to infinity. The purpose of evaluating this limit is to determine if the accumulated area under the curve remains finite. If as \( b \) approaches infinity, the limit is a real, bounded number, then the original integral converges. However, if the limit tends towards infinity, like \[ \lim_{b \to \infty} \ln(b+1) \rightarrow \infty \],the integral is divergent since it indicates unbounded growth.
Evaluating limits, therefore, allows us to systematically and mathematically probe the behavior at infinity to assess convergence or divergence.
This formulation lets us investigate what happens to the integral as the boundary extends to infinity. The purpose of evaluating this limit is to determine if the accumulated area under the curve remains finite. If as \( b \) approaches infinity, the limit is a real, bounded number, then the original integral converges. However, if the limit tends towards infinity, like \[ \lim_{b \to \infty} \ln(b+1) \rightarrow \infty \],the integral is divergent since it indicates unbounded growth.
Evaluating limits, therefore, allows us to systematically and mathematically probe the behavior at infinity to assess convergence or divergence.
Antiderivative Calculation
Finding the antiderivative, or the indefinite integral, is a fundamental operation when working with definite integrals, including improper ones. It involves reversing the process of differentiation to determine a function that, when differentiated, yields the original function.
In the given integral \[ \int_{0}^{b} \frac{1}{x+1} \, dx \],we identify the integrand's antiderivative, which requires an application of substitution if necessary. For \( \frac{1}{x+1} \),we can let \( u = x + 1 \),which simplifies the integrand to \( \frac{1}{u} \) whose antiderivative is \( \ln|u| \).Reversing the substitution, the integral becomes \( \ln|x+1| \).
Calculating these antiderivatives allows us to evaluate definite integrals by the Fundamental Theorem of Calculus. This particular result guides us from finding the antiderivative to plugging in limits to see distance or area over an interval. Hence, antiderivative calculation is the bridge towards evaluating definite integrals and further exploring their convergence.
In the given integral \[ \int_{0}^{b} \frac{1}{x+1} \, dx \],we identify the integrand's antiderivative, which requires an application of substitution if necessary. For \( \frac{1}{x+1} \),we can let \( u = x + 1 \),which simplifies the integrand to \( \frac{1}{u} \) whose antiderivative is \( \ln|u| \).Reversing the substitution, the integral becomes \( \ln|x+1| \).
Calculating these antiderivatives allows us to evaluate definite integrals by the Fundamental Theorem of Calculus. This particular result guides us from finding the antiderivative to plugging in limits to see distance or area over an interval. Hence, antiderivative calculation is the bridge towards evaluating definite integrals and further exploring their convergence.
Other exercises in this chapter
Problem 17
(a) Find the Taylor polynomial of degree 3 about \(x=0\) for \(f(x)=\sin x\). (b) Use your result in (a) to explain why $$ \lim _{x \rightarrow 0} \frac{\sin x}
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Use a spreadsheet to approximate each of the following integrals using the midpoint rule with each of the specified values of \(n .\) \(\int_{0}^{1} x^{2} d x\)
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In Problems 1-30, use integration by parts to evaluate each integral. $$ \int_{0}^{\pi / 3} x \sin x d x $$
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Write out the partial-fraction decomposition of the function \(f(x)\). $$ f(x)=\frac{2 x-3}{x(x+1)} $$
View solution