Problem 22 - Calculus and its applications

Question

Determine whether each improper integral is convergent or divergent, and find its value if it is convergent. $$ \int_{1}^{\infty} \frac{2 t}{t^{2}+1} d t $$

Step-by-Step Solution

Answer

The integral is divergent.

Step 1: Interpret the Integral

Determine the nature of the given integral \( \int_{1}^{\infty} \frac{2t}{t^2+1} \, dt \), which is an improper integral due to its upper limit being infinity.

Step 2: Setup the Limit

Since this is an improper integral with an infinite upper limit, we set up a limit to evaluate it: \[\lim_{b \to \infty} \int_{1}^{b} \frac{2t}{t^2+1} \, dt\]

Step 3: Substitution Method

To evaluate the integral, use the substitution method. Let \( u = t^2 + 1 \), then \( du = 2t \, dt \). This simplifies the integral to \[\int \frac{du}{u}\].

Step 4: Integrate the Simplified Function

The integral \( \int \frac{du}{u} \) is a standard integral, resulting in:\[ \ln |u| + C\]. Since \( u = t^2 + 1 \), substitute back to reach \[ \ln |t^2 + 1|\].

Step 5: Apply Limit to the Antiderivative

Apply the limit to find the value of the improper integral:\[\lim_{b \to \infty} \left[ \ln(b^2+1) - \ln(1^2+1) \right]\].

Step 6: Evaluate the Limit

As \( b \to \infty \), \( \ln(b^2 + 1) \to \infty \). However, since we're evaluating an integral, we need to check if the expressions cancel out. Plugging in, it becomes:\[\lim_{b \to \infty} \ln(b^2 + 1) - \ln 2\] which simplifies to\[\infty - \ln 2 = \infty\].

Step 7: Conclusion

Since the evaluation of the limit goes to infinity, the improper integral \(\int_{1}^{\infty} \frac{2t}{t^2+1} \, dt\) is divergent.

Key Concepts

Convergence and Divergence

Improper integrals are a special type of integral where either the interval is infinite or the function being integrated has an infinite discontinuity. When evaluating these integrals, one of the primary tasks is to determine if they are convergent or divergent.
Convergence implies that the integral has a finite value, often interpreted as finding the area under a curve that extends to infinity or near a point of discontinuity. In contrast, if an integral is divergent, this means that the area under the curve is infinite, indicating the function does not approach a finite limit.
To determine convergence, we commonly use limits. By transforming the improper integral into a limit problem, we can check the behavior of the function as it approaches its boundary conditions. In our exercise, because the limit evaluation resulted in an infinite value, it was concluded that the integral diverges.

Substitution Method

The substitution method is a powerful technique in calculus used to simplify the integration process. It is essentially the reverse of the chain rule used in differentiation.
When we use substitution, we change the variable of integration to something that makes the integral easier to evaluate. Here, we replaced the complex part of the integral, specifically the function inside the integrand, with a single variable.
Steps in Substitution:

• Identify a substitution that simplifies the integral, often a part of the integrand.
• Redefine the integral in terms of the new variable.
• Calculate the differential of the new variable.
• integrate the simpler integral in terms of this new variable.
• Finally, substitute back to the original variable.

In the exercise, we set \( u = t^2 + 1 \), which allowed us to transform the integral into a straightforward logarithmic form of \( \int \frac{du}{u} \). This simplification helps in finding solutions to even more complex integrals.

Limits in Calculus

Limits are fundamental in calculus, especially when dealing with improper integrals. They help evaluate the behavior of a function as it approaches a particular point or infinity.
When computing an improper integral that extends to infinity, we essentially use limits to "stop" the integral at a finite point. This helps us predict the behavior of the integral if it were continued indefinitely.
Calculating Limits:

• Identify the variable approaching a limit, such as \( b \to \infty \).
• Replace this variable in the integral's domain with this limiting behavior.
• Solve the resulting expression using known limit properties.

In our example, we used the limit notation \( \lim_{b \to \infty} \int_{1}^{b} \frac{2t}{t^2+1} \, dt \) to analyze the function's behavior as the upper bound became very large. This method revealed that the integral was divergent because the final limit evaluation resulted in infinity.

Integration Techniques

Integration techniques are various strategies used to solve different types of integrals. While simple integrals can be straightforward, many integrals, especially those involving polynomial fractions or functions squared, require more advanced techniques.
Here are some common techniques:

• Substitution: Useful for transforming the integrand into a simpler form, as seen in the exercise.
• By Parts: This method uses the product rule in reverse, helpful for products of functions.
• Partial Fractions: A technique for breaking down complex rational expressions.
• Trigonometric Integrals and Substitutions: Useful for integrating trig functions or expressions involving them.

In our scenario, after identifying the integral as improper, substitution transformed it into a form amenable to simpler integration (into \( \int \frac{du}{u} \)). This specific technique relied on identifying the quadratic structure \( t^2 + 1 \) and using it to simplify integration, thereby demonstrating the power of choosing the right technique for solving integrals.