Chapter 8: Problem 10
Evaluate each improper integral or show that it diverges. \(\int_{1}^{\infty} \frac{x}{\left(1+x^{2}\right)^{2}} d x\)
Short Answer
Expert verified
The integral converges to \( \frac{1}{2} \).
Step by step solution
01
Determine if the Integral is Improper
The given integral \( \int_{1}^{\infty} \frac{x}{(1+x^2)^2} \, dx \) is improper because the upper limit of integration is infinity.
02
Set up a Limit to Evaluate the Improper Integral
To handle the improper integral, we set it up as a limit: \[ \lim_{b \to \infty} \int_{1}^{b} \frac{x}{(1+x^2)^2} \, dx. \]
03
Use Substitution to Solve the Integral
Let \( u = 1 + x^2 \), then \( du = 2x \, dx \). This implies that \( x \, dx = \frac{1}{2} du \). The integral becomes: \[ \frac{1}{2} \int \frac{1}{u^2} \, du. \]
04
Integrate Using Power Rule for Integrals
The integral \( \int \frac{1}{u^2} \, du \) can be integrated using the power rule: \[ \int u^{-2} \, du = -u^{-1} + C, \] resulting in: \[ -\frac{1}{u} + C. \]
05
Back-substitute for \(u\)
Substitute back \( u = 1 + x^2 \), which gives \[ -\frac{1}{1+x^2} + C. \]
06
Evaluate the Definite Integral
Now consider the limits: \[ \lim_{b \to \infty} \left( -\frac{1}{1+b^2} \right) - \left( -\frac{1}{1+1^2} \right). \]
07
Evaluate the Limit
Evaluate the expression: \[ \lim_{b \to \infty} -\frac{1}{1+b^2} \] tends to 0, because as \(b\) approaches infinity, \(1+b^2\) becomes very large. The term \( -\frac{1}{2} \) remains constant.
08
Conclusion
Finally, we obtain \[ 0 - (-\frac{1}{2}) = \frac{1}{2}. \] Thus, the integral converges to \( \frac{1}{2} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Integration Techniques
When evaluating integrals, especially improper integrals like \( \int_{1}^{\infty} \frac{x}{(1+x^2)^2} \, dx \), selecting the right technique is crucial. One common technique is substitution. It simplifies integrands and helps solve complex integrals. This method involves selecting a part of the integrand to replace with a simpler variable, often labeled as \( u \), to transform the integral into a more manageable form.
As you substitute, ensure the differential \( dx \) changes accordingly. In this exercise, by setting \( u = 1 + x^2 \), the derivative becomes \( du = 2x \, dx \). Hence, we rearranged \( x \, dx \) to substitute it in terms of \( du \), which is \( \frac{1}{2} du \). This step smoothens the integration process.
Regardless of the technique used, always revert back to the original variable by substituting back once the integration is complete. This ensures that you arrive at the solution in the terms given in the original problem.
As you substitute, ensure the differential \( dx \) changes accordingly. In this exercise, by setting \( u = 1 + x^2 \), the derivative becomes \( du = 2x \, dx \). Hence, we rearranged \( x \, dx \) to substitute it in terms of \( du \), which is \( \frac{1}{2} du \). This step smoothens the integration process.
Regardless of the technique used, always revert back to the original variable by substituting back once the integration is complete. This ensures that you arrive at the solution in the terms given in the original problem.
Convergence of Integrals
Determining whether an improper integral converges or diverges is vital. Convergence means the integral approaches a specific value, while divergence indicates it tends to infinity or does not settle on a particular value.
To check convergence, especially when dealing with infinity in the integral, setting up a limit is often used. For the given integral \( \int_{1}^{\infty} \frac{x}{(1+x^2)^2} \, dx \), we replaced the upper limit with a parameter \( b \), creating \( \lim_{b \to \infty} \int_{1}^{b} \frac{x}{(1+x^2)^2} \, dx \).
After solving the integral, evaluating the limit helps determine if the integral's value stabilizes (converges) as \( b \) approaches infinity. In this exercise, the final evaluation led to a finite number, \( \frac{1}{2} \), confirming convergence.
To check convergence, especially when dealing with infinity in the integral, setting up a limit is often used. For the given integral \( \int_{1}^{\infty} \frac{x}{(1+x^2)^2} \, dx \), we replaced the upper limit with a parameter \( b \), creating \( \lim_{b \to \infty} \int_{1}^{b} \frac{x}{(1+x^2)^2} \, dx \).
After solving the integral, evaluating the limit helps determine if the integral's value stabilizes (converges) as \( b \) approaches infinity. In this exercise, the final evaluation led to a finite number, \( \frac{1}{2} \), confirming convergence.
Limit Evaluations
Evaluating limits is a cornerstone in understanding improper integrals. It connects infinite boundaries within an integrand to a finite solution. When you encounter an integral extending to infinity, replace the infinity with a limit variable like \( b \), then take \( b \rightarrow \infty \) after integrating.
In our problem, the limit at \( b \to \infty \) shows how the fraction \( -\frac{1}{1+b^2} \) affects the final calculation. As \( b \) gets larger, the denominator \( 1 + b^2 \) grows swiftly, meaning the whole expression approaches zero. This behavior is crucial since it determines the convergence of our integral.
By analyzing the limit, we confirmed the contributory effect of the infinite boundary diminishes over distance. Combine such an approach with substitution and other techniques for successful evaluations of improper integrals.
In our problem, the limit at \( b \to \infty \) shows how the fraction \( -\frac{1}{1+b^2} \) affects the final calculation. As \( b \) gets larger, the denominator \( 1 + b^2 \) grows swiftly, meaning the whole expression approaches zero. This behavior is crucial since it determines the convergence of our integral.
By analyzing the limit, we confirmed the contributory effect of the infinite boundary diminishes over distance. Combine such an approach with substitution and other techniques for successful evaluations of improper integrals.
Substitution Method
The substitution method is a powerful tool for integrals, particularly useful when faced with algebraic expressions that are tricky to integrate directly. It involves replacing a part of the integrand with a new variable, simplifying the process.
In this exercise, we applied substitution by letting \( u = 1 + x^2 \). This replacement simplifies the integral \( \int \frac{x}{(1+x^2)^2} \, dx \) into a form that is easier to manage: \( \frac{1}{2} \int \frac{1}{u^2} \, du \). This operation turns the original problem into a straightforward algebraic integration using known rules like the power rule.
Always remember, the final step after integrating with respect to \( u \) involves reversing the substitution. This returns the solution in terms of the original variable, completing the integration process in consistency with the initial problem. This ensures clarity and correctness when handling final answers.
In this exercise, we applied substitution by letting \( u = 1 + x^2 \). This replacement simplifies the integral \( \int \frac{x}{(1+x^2)^2} \, dx \) into a form that is easier to manage: \( \frac{1}{2} \int \frac{1}{u^2} \, du \). This operation turns the original problem into a straightforward algebraic integration using known rules like the power rule.
Always remember, the final step after integrating with respect to \( u \) involves reversing the substitution. This returns the solution in terms of the original variable, completing the integration process in consistency with the initial problem. This ensures clarity and correctness when handling final answers.
