Solving The Improper Integral ∫[0 To Π/2] (e^(x^2)-1)/(sin^2(x)) Dx

Introduction

The challenge of evaluating definite integrals, especially improper integrals, often presents a fascinating journey through the realms of calculus. Among these, certain integrals stand out due to their intricate nature and the necessity for clever techniques to arrive at a solution. This article delves into the process of solving the intriguing improper integral:

$$\int_0^{\frac{\pi}{2}} \frac{e^{x^2}-1}{\sin^2(x)}dx$$

This integral, with its combination of exponential and trigonometric functions, requires a delicate approach, blending various integration strategies and a keen understanding of limit behavior near singularities. It is a quintessential example of why integral calculus is both a powerful tool and an art form. The journey to its solution will illuminate key concepts in advanced calculus and highlight the beauty of mathematical problem-solving.

Identifying the Challenge: Why is it "Improper"?

Before diving into the solution, it's crucial to recognize why this integral is classified as "improper." The term improper integral arises primarily due to two reasons:

1. Infinite Limits of Integration: One or both limits of integration extend to infinity.
2. Discontinuities in the Integrand: The function being integrated (the integrand) has one or more discontinuities within the interval of integration.

In our case, the integral

$$\int_0^{\frac{\pi}{2}} \frac{e^{x^2}-1}{\sin^2(x)}dx$$

falls into the second category. While the limits of integration are finite (0 and π/2), the integrand exhibits a discontinuity at x = 0. This is because as x approaches 0, sin(x) also approaches 0, causing the denominator sin²(x) to approach 0, and thus, the entire fraction to approach infinity. This singularity at x = 0 makes the integral "improper." To handle this, we must employ techniques that carefully consider the behavior of the integrand near this point.

To illustrate, consider the limit:

$$\lim_{x \to 0} \frac{e^{x^2}-1}{\sin^2(x)}$$

Applying L'Hôpital's Rule (since we have an indeterminate form of type 0/0) twice, we find:

First application:

$$\lim_{x \to 0} \frac{2xe^{x^2}}{2\sin(x)\cos(x)} = \lim_{x \to 0} \frac{xe^{x^2}}{\sin(x)\cos(x)}$$

Second application (after rewriting as a product and applying L'Hôpital's Rule again):

$$\lim_{x \to 0} \frac{e^{x^2} + 2x^2e^{x^2}}{\cos^2(x) - \sin^2(x)} = \frac{1}{1} = 1$$

Despite this limit existing, the singularity still requires careful treatment. This involves reformulating the integral as a limit of proper integrals, which we will explore in the next section.

The Strategy: Transforming the Improper Integral

To properly evaluate the improper integral, the initial step involves transforming it into a form that addresses the singularity at x = 0. This is achieved by introducing a limit that approaches the point of discontinuity. The original integral:

$$\int_0^{\frac{\pi}{2}} \frac{e^{x^2}-1}{\sin^2(x)}dx$$

is rewritten as a limit:

$$\lim_{a \to 0^+} \int_a^{\frac{\pi}{2}} \frac{e^{x^2}-1}{\sin^2(x)}dx$$

Here, a represents a value approaching 0 from the positive side (denoted by 0+). This transformation allows us to work with a proper integral (from a to π/2) and then examine the behavior as a gets arbitrarily close to 0. This method is the standard approach for dealing with singularities at the limits of integration. By introducing this limit, we circumvent the direct evaluation at the point of discontinuity and instead analyze the integral's trend as we approach it.

Now, the challenge shifts to evaluating the integral:

$$\int_a^{\frac{\pi}{2}} \frac{e^{x^2}-1}{\sin^2(x)}dx$$

This integral, although proper, is still not straightforward. The combination of the exponential term e^(x2) and the trigonometric term sin²(x) suggests that direct integration might be difficult. This is where the technique of integration by parts comes into play. Integration by parts is a powerful tool for handling integrals involving products of functions, and it is particularly useful when one part of the product can be simplified through differentiation.

In the following sections, we will apply integration by parts, carefully choosing which part of the integrand to differentiate and which to integrate. This step is crucial in simplifying the integral and making it more amenable to evaluation. The choice of parts is guided by the goal of reducing the complexity of the integral, often by transforming it into a form that is either directly integrable or can be further simplified.

Integration by Parts: A Crucial Step

The technique of integration by parts is pivotal in tackling the integral

$$\int_a^{\frac{\pi}{2}} \frac{e^{x^2}-1}{\sin^2(x)}dx$$

This method is based on the product rule for differentiation and is mathematically expressed as:

$$\int u dv = uv - \int v du$$

where u and v are functions of x, and du and dv are their respective differentials. The key to successfully applying integration by parts lies in the strategic selection of u and dv. The goal is to choose these parts such that the resulting integral on the right-hand side, ∫v du, is simpler to evaluate than the original integral, ∫u dv.

For our integral, a judicious choice for u and dv is:

• u = e^(x2) - 1
• dv = 1/sin²(x) dx

This selection is motivated by the fact that the derivative of e^(x2) - 1 is 2xe^(x2), which, while not simpler in isolation, combines well with other terms that will arise. The antiderivative of 1/sin²(x) is -cot(x), which is a standard result and manageable. Following these choices, we find:

• du = 2xe^(x2) dx
• v = -cot(x)

Applying the integration by parts formula, we get:

$$\int_a^{\frac{\pi}{2}} \frac{e^{x^2}-1}{\sin^2(x)}dx = \left[-(e^{x^2}-1)\cot(x)\right]_a^{\frac{\pi}{2}} + \int_a^{\frac{\pi}{2}} 2xe^{x^2}\cot(x) dx$$

Let's analyze the terms that arise from this application. The first term,

$$\left[-(e^{x^2}-1)\cot(x)\right]_a^{\frac{\pi}{2}}$$

is an evaluation of the function -(e^(x2)-1)cot(x) at the limits of integration, π/2 and a. This term will contribute a specific value, which we will compute later. The second term,

$$\int_a^{\frac{\pi}{2}} 2xe^{x^2}\cot(x) dx$$

is a new integral. While it might not appear immediately simpler, it is a crucial step forward. This new integral involves the product of 2xe^(x2) and cot(x), which, although still complex, is more manageable than the original integrand. Specifically, the exponential term e^(x2) is now accompanied by x, which will facilitate further integration. To handle this new integral, we will again employ integration by parts, demonstrating the power of this technique in simplifying complex integrals iteratively.

Iterating Integration by Parts: Taming the New Integral

After the first application of integration by parts, we arrived at the integral

$$\int_a^{\frac{\pi}{2}} 2xe^{x^2}\cot(x) dx$$

To further simplify this, we apply integration by parts once more. This iterative approach is common in dealing with integrals that don't yield to a single application of the technique. For this second round, we make the following choices for u and dv:

• u = cot(x)
• dv = 2xe^(x2) dx

These choices are strategic. By selecting cot(x) as u, we simplify it upon differentiation, as its derivative is -csc²(x). Choosing 2xe^(x2) dx as dv allows for direct integration, yielding e^(x2) as v. Following these choices, we find:

• du = -csc²(x) dx
• v = e^(x2)

Applying the integration by parts formula again, we obtain:

$$\int_a^{\frac{\pi}{2}} 2xe^{x^2}\cot(x) dx = \left[e^{x^2}\cot(x)\right]_a^{\frac{\pi}{2}} + \int_a^{\frac{\pi}{2}} e^{x^2}\csc^2(x) dx$$

Now, substituting this back into our original equation from the first integration by parts, we have:

$$\int_a^{\frac{\pi}{2}} \frac{e^{x^2}-1}{\sin^2(x)}dx = \left[-(e^{x^2}-1)\cot(x)\right]_a^{\frac{\pi}{2}} + \left[e^{x^2}\cot(x)\right]_a^{\frac{\pi}{2}} + \int_a^{\frac{\pi}{2}} e^{x^2}\csc^2(x) dx$$

Notice a remarkable simplification: the terms involving cot(x) evaluated at the limits π/2 and a partially cancel each other. Specifically, the term e^(x2)cot(x) appears with opposite signs, leading to cancellation. This cancellation is a significant step forward, as it reduces the complexity of the expression and brings us closer to a manageable form. After the cancellation, we are left with:

$$\int_a^{\frac{\pi}{2}} \frac{e^{x^2}-1}{\sin^2(x)}dx = \left[\cot(x)\right]_a^{\frac{\pi}{2}} + \int_a^{\frac{\pi}{2}} e^{x^2}\csc^2(x) dx$$

This expression is considerably simpler than our starting point. We now have a basic trigonometric term evaluated at the limits and a new integral involving e^(x2) and csc²(x). The integral on the right-hand side appears daunting, but we can rewrite $csc^2(x)$ as $1 + cot^2(x)$. This seemingly small step paves the way for further simplification and eventual evaluation of the integral.

Rewriting and Recognizing a Pattern

Following the iterative integration by parts and the resulting simplifications, we have arrived at:

$$\int_a^{\frac{\pi}{2}} \frac{e^{x^2}-1}{\sin^2(x)}dx = \left[\cot(x)\right]_a^{\frac{\pi}{2}} + \int_a^{\frac{\pi}{2}} e^{x^2}\csc^2(x) dx$$

As highlighted earlier, rewriting csc²(x) as 1 + cot²(x) is a crucial step. This identity allows us to express the integral in a form that reveals a pattern and facilitates further simplification. Substituting this identity into the integral, we get:

$$\int_a^{\frac{\pi}{2}} \frac{e^{x^2}-1}{\sin^2(x)}dx = \left[\cot(x)\right]_a^{\frac{\pi}{2}} + \int_a^{\frac{\pi}{2}} e^{x^2}(1 + \cot^2(x)) dx$$

Expanding the integral, we have:

$$\int_a^{\frac{\pi}{2}} \frac{e^{x^2}-1}{\sin^2(x)}dx = \left[\cot(x)\right]_a^{\frac{\pi}{2}} + \int_a^{\frac{\pi}{2}} e^{x^2} dx + \int_a^{\frac{\pi}{2}} e^{x^2}\cot^2(x) dx$$

Now, let's focus on the term ∫[a to π/2] e^(x2)cot²(x) dx. Recalling that cot(x) = cos(x)/sin(x), we can rewrite this term as:

$$\int_a^{\frac{\pi}{2}} e^{x^2}\frac{\cos^2(x)}{\sin^2(x)} dx$$

Using the identity cos²(x) = 1 - sin²(x), we can further rewrite the integral as:

$$\int_a^{\frac{\pi}{2}} e^{x^2}\frac{1 - \sin^2(x)}{\sin^2(x)} dx = \int_a^{\frac{\pi}{2}} e^{x^2}(\csc^2(x) - 1) dx$$

Separating this into two integrals gives us:

$$\int_a^{\frac{\pi}{2}} e^{x^2}\csc^2(x) dx - \int_a^{\frac{\pi}{2}} e^{x^2} dx$$

Substituting this back into our main equation, we observe a remarkable cancellation:

$$\int_a^{\frac{\pi}{2}} \frac{e^{x^2}-1}{\sin^2(x)}dx = \left[\cot(x)\right]_a^{\frac{\pi}{2}} + \int_a^{\frac{\pi}{2}} e^{x^2} dx + \int_a^{\frac{\pi}{2}} e^{x^2}\csc^2(x) dx - \int_a^{\frac{\pi}{2}} e^{x^2} dx$$

The integrals ∫[a to π/2] e^(x2) dx cancel each other out, leaving us with a significantly simpler expression:

$$\int_a^{\frac{\pi}{2}} \frac{e^{x^2}-1}{\sin^2(x)}dx = \left[\cot(x)\right]_a^{\frac{\pi}{2}} + \int_a^{\frac{\pi}{2}} e^{x^2}\csc^2(x) dx$$

This simplification is a testament to the power of strategic manipulation and the recognition of patterns in integral calculus. At this juncture, our integral has been reduced to manageable terms, setting the stage for the final evaluation by taking the limit as a approaches 0.

Final Evaluation: Taking the Limit

Having simplified the integral to

$$\int_a^{\frac{\pi}{2}} \frac{e^{x^2}-1}{\sin^2(x)}dx = [-(e^{x^2}-1)cot(x)]_a^{\frac{\pi}{2}} + [e^{x^2}cot(x)]_a^{\frac{\pi}{2}}$$

which further simplifies to

$$\int_a^{\frac{\pi}{2}} \frac{e^{x^2}-1}{\sin^2(x)}dx = [cot(x)]_a^{\frac{\pi}{2}} + \int_a^{\frac{\pi}{2}} e^{x^2} csc^2(x)dx$$

After rewriting $csc^2(x)$ and applying trigonometric identities, we have now arrived at the simplified expression:

$$\int_a^{\frac{\pi}{2}} \frac{e^{x^2}-1}{\sin^2(x)}dx = \left[\cot(x)\right]_a^{\frac{\pi}{2}}$$

Now, we need to evaluate the term cot(x) at the limits of integration, π/2 and a. Recall that cot(x) = cos(x)/sin(x). At x = π/2, cot(π/2) = cos(π/2)/sin(π/2) = 0/1 = 0. Thus, the evaluation at the upper limit is straightforward.

However, at the lower limit x = a, where a approaches 0, we have cot(a) = cos(a)/sin(a). As a approaches 0, cos(a) approaches 1, and sin(a) approaches 0. Thus, cot(a) approaches infinity. This is a critical point that requires careful handling when taking the limit.

To properly evaluate the improper integral, we need to find:

$$\lim_{a \to 0^+} \left(\cot(\frac{\pi}{2}) - \cot(a)\right) = \lim_{a \to 0^+} (0 - \cot(a)) = -\lim_{a \to 0^+} \cot(a)$$

Since cot(a) approaches infinity as a approaches 0 from the positive side, the limit is negative infinity. Therefore, we have:

$$\lim_{a \to 0^+} \int_a^{\frac{\pi}{2}} \frac{e^{x^2}-1}{\sin^2(x)}dx = -\lim_{a \to 0^+} \cot(a) = -\infty$$

This result indicates that the improper integral diverges. The function

$$\frac{e^{x^2}-1}{\sin^2(x)}$$

grows without bound near x = 0, causing the integral to not converge to a finite value. This divergence underscores the importance of carefully analyzing the behavior of integrands near singularities and employing appropriate techniques to handle improper integrals.

Conclusion: A Journey Through Integral Calculus

The evaluation of the improper integral

$$\int_0^{\frac{\pi}{2}} \frac{e^{x^2}-1}{\sin^2(x)}dx$$

proved to be a comprehensive journey through several key concepts and techniques in integral calculus. Initially identified as an improper integral due to the singularity at x = 0, it required a transformation into a limit of proper integrals. The application of integration by parts, performed iteratively, was crucial in simplifying the integrand. Strategic choices of u and dv, guided by the goal of reducing complexity, led to significant cancellations and the emergence of recognizable patterns.

The rewriting of trigonometric terms, particularly the identity csc²(x) = 1 + cot²(x), played a pivotal role in uncovering cancellations and further simplifying the expression. The eventual cancellation of integrals involving e^(x2) highlighted the power of algebraic manipulation in conjunction with calculus techniques.

Finally, the evaluation of the limit as a approached 0 revealed that the integral diverges. This outcome underscores the importance of careful analysis when dealing with improper integrals, as not all such integrals converge to a finite value. The divergence in this case is attributed to the unbounded growth of the integrand near the singularity.

In conclusion, this problem serves as an excellent example of the depth and richness of integral calculus. It demonstrates the necessity for a multifaceted approach, combining techniques such as integration by parts, trigonometric identities, and limit evaluations. The process of solving this integral not only reinforces fundamental skills but also provides insights into the behavior of functions and the subtleties of improper integrals. It is a testament to the power and beauty of mathematical problem-solving.