Solving The Integral Of E^x Sin X A Step-by-Step Guide

Jul 14, 2025 by ADMIN 55 views

Unveiling the Integral of e^x sin x A Comprehensive Guide

The integral ∫ e^x sin x dx is a classic example in calculus that demonstrates the power and elegance of integration techniques. This integral, involving the product of an exponential function (e^x) and a trigonometric function (sin x), doesn't succumb to simple substitution or direct application of standard integration formulas. Instead, it necessitates the clever use of integration by parts, a method that elegantly transforms the integral into a more manageable form. In this comprehensive guide, we will embark on a step-by-step journey to unravel the intricacies of this integral, exploring the underlying concepts and mastering the techniques required to solve it. We will not only delve into the mechanics of the solution but also illuminate the conceptual understanding that makes this problem a cornerstone in the study of integral calculus. This exploration will involve multiple applications of integration by parts, strategic manipulations, and ultimately, an elegant algebraic resolution. By the end of this guide, you will not only be able to confidently evaluate this specific integral but also gain a deeper appreciation for the art of integration and the versatility of integration by parts. Moreover, we will discuss the importance of this integral in various fields of science and engineering, highlighting its practical applications and theoretical significance. This integral serves as a building block for more complex problems and provides a fundamental understanding of how different types of functions interact under integration.

Before we dive into the solution, let's refresh our understanding of integration by parts. Integration by parts is a powerful technique derived from the product rule of differentiation. The product rule states that the derivative of the product of two functions, u(x) and v(x), is given by:

d/ dx [u(x) v(x)] = u'(x) v(x) + u(x) v'(x)

Integrating both sides with respect to x, we get:

∫ d/ dx [u(x) v(x)] dx = ∫ u'(x) v(x) dx + ∫ u(x) v'(x) dx

u(x) v(x) = ∫ v(x) u'(x) dx + ∫ u(x) v'(x) dx

Rearranging this equation, we arrive at the integration by parts formula:

∫ u dv = uv - ∫ 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 choice of u and dv. The goal is to select u and dv such that the integral ∫ v du is simpler to evaluate than the original integral ∫ u dv. A helpful mnemonic for choosing u is LIATE, which stands for Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, and Exponential. This acronym suggests a general order of preference for choosing u: if you have a logarithmic function, choose that as u; if you have an inverse trigonometric function, choose that; and so on. However, LIATE is just a guideline, and sometimes you may need to deviate from it to find the most effective approach. The success of integration by parts often hinges on making the right choice for u and dv, and this often comes with practice and experience. Understanding the underlying principle of integration by parts, which stems from the product rule of differentiation, is crucial for mastering this technique. By carefully selecting u and dv, we can transform complex integrals into simpler ones, making the process of integration more manageable and efficient. In the case of ∫ e^x sin x dx, we will see how this technique is applied iteratively to arrive at the solution.

Now, let's tackle the integral ∫ e^x sin x dx using integration by parts. This integral is a prime example where the technique needs to be applied twice to reach a solution. Here's the breakdown:

Step 1: First Application of Integration by Parts

Following the LIATE rule, we choose:

u = sin x (Trigonometric function)
dv = e^x dx (Exponential function)

Then, we find:

du = cos x dx
v = ∫ e^x dx = e^x

Applying the integration by parts formula, ∫ u dv = uv - ∫ v du, we get:

∫ e^x sin x dx = sin x e^x - ∫ e^x cos x dx

We now have a new integral, ∫ e^x cos x dx, which also requires integration by parts.

Step 2: Second Application of Integration by Parts

For the integral ∫ e^x cos x dx, we again apply integration by parts. This time, we choose:

u = cos x (Trigonometric function)
dv = e^x dx (Exponential function)

Then, we find:

du = -sin x dx
v = ∫ e^x dx = e^x

Applying the integration by parts formula again, we get:

∫ e^x cos x dx = cos x e^x - ∫ e^x (-sin x) dx

∫ e^x cos x dx = cos x e^x + ∫ e^x sin x dx

Step 3: Combining and Solving for the Integral

Now, we substitute the result from Step 2 back into the equation from Step 1:

∫ e^x sin x dx = sin x e^x - (cos x e^x + ∫ e^x sin x dx)

∫ e^x sin x dx = sin x e^x - cos x e^x - ∫ e^x sin x dx

Notice that the original integral, ∫ e^x sin x dx, appears on both sides of the equation. This is a crucial observation. We can now treat the integral as an algebraic variable and solve for it.

Let I = ∫ e^x sin x dx. Then the equation becomes:

I = sin x e^x - cos x e^x - I

Adding I to both sides, we get:

2I = sin x e^x - cos x e^x

Dividing both sides by 2, we find:

I = (1/2) e^x (sin x - cos x)

Step 4: Adding the Constant of Integration

Finally, we add the constant of integration, C, to obtain the complete solution:

∫ e^x sin x dx = (1/2) e^x (sin x - cos x) + C

Thus, we have successfully evaluated the integral ∫ e^x sin x dx using integration by parts. This process demonstrates the power and elegance of the technique, especially when applied iteratively. The key is to strategically choose u and dv and to recognize when the original integral reappears, allowing for an algebraic solution.

Another elegant approach to solving ∫ e^x sin x dx involves the use of complex exponentials. This method leverages Euler's formula, which connects complex exponentials with trigonometric functions. Euler's formula states:

e^(ix) = cos x + i sin x

Where i is the imaginary unit (i^2 = -1). We can isolate sin x from this formula by taking the imaginary part:

sin x = Im(e^(ix))

Now, we can rewrite the original integral as:

∫ e^x sin x dx = ∫ e^x Im(e^(ix)) dx = Im(∫ e^x e^(ix) dx)

This allows us to work with the integral:

∫ e^x e^(ix) dx = ∫ e^((1+i)x) dx

This integral is straightforward to evaluate:

∫ e^((1+i)x) dx = e^((1+i)x) / (1 + i) + C

To simplify the denominator, we multiply the numerator and denominator by the complex conjugate of (1 + i), which is (1 - i):

e^((1+i)x) / (1 + i) = e^((1+i)x) (1 - i) / ((1 + i)(1 - i))

e^((1+i)x) / (1 + i) = e^((1+i)x) (1 - i) / (1 - i^2)

Since i^2 = -1:

e^((1+i)x) / (1 + i) = e^((1+i)x) (1 - i) / 2

Now, we substitute e^((1+i)x) with e^x e^(ix) and apply Euler's formula again:

e^((1+i)x) (1 - i) / 2 = e^x (cos x + i sin x) (1 - i) / 2

Expanding the expression:

e^x (cos x + i sin x) (1 - i) / 2 = e^x (cos x - i cos x + i sin x - i^2 sin x) / 2

e^x (cos x - i cos x + i sin x + sin x) / 2 = e^x [(cos x + sin x) + i(sin x - cos x)] / 2

Now, we take the imaginary part of this expression:

Im(e^x [(cos x + sin x) + i(sin x - cos x)] / 2) = e^x (sin x - cos x) / 2

Finally, we add the constant of integration:

∫ e^x sin x dx = (1/2) e^x (sin x - cos x) + C

This result matches the solution we obtained using integration by parts. The complex exponential method offers a different perspective on the problem and demonstrates the power of complex analysis in solving real-valued integrals. This approach can be particularly useful for integrals involving products of exponential and trigonometric functions, as it streamlines the process by converting trigonometric functions into complex exponentials, which are often easier to manipulate.

The integral ∫ e^x sin x dx is not just a mathematical curiosity; it has significant practical applications in various fields of science and engineering. This integral, and others like it involving products of exponential and trigonometric functions, arises frequently in the analysis of systems that exhibit damped oscillatory behavior. Damped oscillations are ubiquitous in the physical world, appearing in mechanical systems, electrical circuits, and even biological systems.

In physics, this type of integral is encountered when analyzing damped harmonic oscillators. A damped harmonic oscillator is a system that oscillates with decreasing amplitude over time due to energy dissipation, such as friction or air resistance. Examples include a pendulum swinging with air resistance, a mass-spring system with a damper, and the oscillations in an RLC circuit. The solution to the equations of motion for these systems often involves terms of the form e^(-αt) sin(ωt) or e^(-αt) cos(ωt), where α is the damping coefficient and ω is the angular frequency. The integral ∫ e^x sin x dx provides a fundamental building block for analyzing the energy dissipation and the time-dependent behavior of these systems.

In electrical engineering, this integral appears in the analysis of alternating current (AC) circuits, particularly those containing resistors, inductors, and capacitors (RLC circuits). The behavior of these circuits is governed by differential equations that involve sinusoidal functions multiplied by exponential decay terms. The integral ∫ e^x sin x dx is crucial for determining the transient response of these circuits, which describes how the current and voltage change over time when the circuit is subjected to a sudden change in input, such as switching a circuit on or off. Understanding the transient response is essential for designing stable and efficient electrical systems.

Beyond physics and engineering, integrals of this form also arise in signal processing. Signals that decay exponentially while oscillating are common in many applications, such as audio processing, image processing, and telecommunications. For instance, the impulse response of a linear time-invariant (LTI) system, which describes the system's output when subjected to a brief input pulse, can often be expressed as a sum of exponentially decaying sinusoids. The integral ∫ e^x sin x dx is essential for analyzing and manipulating these signals, allowing engineers to design filters, compress data, and extract information from noisy signals.

Furthermore, this integral is important in advanced mathematical concepts. It serves as a fundamental example in the study of Fourier analysis and Laplace transforms, which are powerful tools for analyzing functions and solving differential equations. Fourier analysis decomposes a function into a sum of sinusoids, while Laplace transforms convert differential equations into algebraic equations, making them easier to solve. The ability to evaluate integrals like ∫ e^x sin x dx is crucial for mastering these advanced techniques.

In summary, the integral ∫ e^x sin x dx is not just a theoretical exercise; it is a practical tool with wide-ranging applications. Its importance stems from its ability to model and analyze systems that exhibit damped oscillatory behavior, which are prevalent in physics, engineering, and other scientific disciplines. Mastering this integral provides a solid foundation for understanding more complex problems and for applying mathematical techniques to real-world phenomena.

In this comprehensive guide, we have thoroughly explored the integral ∫ e^x sin x dx, a cornerstone problem in integral calculus. We began by understanding the necessity of integration by parts, a powerful technique for integrating products of functions. We then meticulously applied integration by parts twice, strategically choosing u and dv to simplify the integral. This process led us to an algebraic equation where the original integral reappeared, allowing us to solve for it directly. We also explored an alternative method using complex exponentials, leveraging Euler's formula to transform the trigonometric function into a complex exponential, streamlining the integration process. Both methods converged to the same elegant solution:

∫ e^x sin x dx = (1/2) e^x (sin x - cos x) + C

Beyond the mechanics of the solution, we emphasized the conceptual understanding that underlies integration by parts and the strategic choices involved in its application. We highlighted the importance of the LIATE mnemonic as a guideline for choosing u but also stressed the need for flexibility and adaptability in selecting the most effective approach. Furthermore, we delved into the practical applications of this integral in various fields, including physics, electrical engineering, and signal processing. We saw how integrals of this form arise in the analysis of damped harmonic oscillators, RLC circuits, and the processing of signals with exponentially decaying sinusoidal components. This exploration underscored the relevance of this integral to real-world phenomena and its role in solving practical problems.

Finally, we noted the significance of this integral in the context of advanced mathematical concepts, such as Fourier analysis and Laplace transforms. Its mastery serves as a crucial stepping stone for understanding these more sophisticated techniques. In conclusion, the integral ∫ e^x sin x dx is not just an exercise in calculus; it is a gateway to a deeper understanding of mathematical principles and their applications. By mastering this integral, you not only enhance your integration skills but also gain valuable insights into the interplay between exponential and trigonometric functions and their role in modeling the world around us. This comprehensive exploration should empower you to confidently tackle similar integration problems and appreciate the beauty and power of calculus.