Solving The Integral Of X E^(x^2) A Step-by-Step Guide

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Introduction

In the realm of calculus, integration is a fundamental operation with widespread applications across various fields of science and engineering. Integrating functions allows us to determine areas under curves, solve differential equations, and model numerous physical phenomena. Among the diverse types of integrals encountered, those involving exponential functions often pose a unique challenge. This article delves into the intricacies of solving the integral ∫xe(x2) dx, a classic example that showcases the power of u-substitution, a crucial technique in integral calculus.

Understanding this integral is essential for students and professionals alike, as it demonstrates how to tackle more complex problems involving exponential functions. The process involves recognizing the composite structure of the integrand and applying the appropriate substitution to simplify the expression. Mastering this technique not only enhances your ability to solve integrals but also deepens your understanding of the underlying principles of calculus.

We will begin by outlining the problem and its context, then meticulously walk through the solution step-by-step. Furthermore, we will discuss the rationale behind each step, providing insights into why the chosen method is effective. By the end of this guide, you will not only be able to solve this particular integral but also gain a broader perspective on how to approach similar problems. Let's embark on this journey to unravel the solution to ∫xe(x2) dx and solidify your understanding of integral calculus.

Problem Statement: ∫xe(x2) dx

The problem at hand is to evaluate the indefinite integral ∫xe(x2) dx. This integral features an exponential function, e(x2), multiplied by a variable term, x. Such integrals are common in various contexts, including physics, engineering, and applied mathematics. The key to solving this integral lies in recognizing the structure of the integrand and applying an appropriate technique. In this case, the method of u-substitution is particularly effective.

Before diving into the solution, it’s crucial to understand why this method is suitable. The integrand consists of a composite function, e(x2), where the inner function is x^2 and its derivative, 2x, is present (up to a constant factor) in the integrand. This presence of a function and its derivative suggests that u-substitution will simplify the integral. The technique of u-substitution is based on the chain rule in differentiation, which states that the derivative of a composite function f(g(x)) is f'(g(x)) * g'(x). By reversing this process, we can often simplify integrals involving composite functions.

To put it simply, u-substitution involves replacing a part of the integrand with a new variable, u, which makes the integral easier to solve. This method is especially useful when the integrand contains a function and its derivative (or a multiple thereof). By carefully choosing the substitution, we can transform the integral into a simpler form that we can readily evaluate. The choice of u is often the most challenging aspect of this technique, but with practice, you can develop an intuition for selecting the appropriate substitution.

In the context of ∫xe(x2) dx, the natural choice for u is x^2 because its derivative, 2x, is present in the integrand (ignoring the constant factor). This observation is the cornerstone of our approach. By substituting u = x^2, we will transform the integral into a more manageable form involving e^u. The subsequent steps will involve differentiating our substitution, expressing the original integral in terms of u, and then evaluating the resulting integral. Finally, we will substitute back to express the result in terms of x. This process may seem complex initially, but with each step, the underlying logic will become clearer. Let's now move on to the solution, where we will detail each step of the u-substitution method.

Solution Using u-Substitution

To solve the integral ∫xe(x2) dx, we will employ the method of u-substitution. This technique is particularly effective when the integrand contains a composite function and its derivative (or a multiple thereof). In this case, the composite function is e(x2), and its derivative is closely related to the x term present in the integrand.

Step 1: Choose the substitution The first step in u-substitution is to identify a suitable substitution. Let's choose:

u = x^2

This choice is motivated by the fact that the derivative of x^2 is 2x, which is present in the integrand (up to a constant factor). Selecting the inner function of the composite function as u is a common and effective strategy.

Step 2: Find du Next, we differentiate u with respect to x to find du:

du/dx = d(x^2)/dx = 2x

From this, we can express du in terms of dx:

du = 2x dx

Step 3: Adjust the integral Now, we need to rewrite the original integral in terms of u. Notice that we have x dx in the integrand, but we have 2x dx in our expression for du. To match these, we can divide both sides of the du equation by 2:

(1/2) du = x dx

Now we can substitute u and (1/2) du into the original integral:

∫xe(x2) dx = ∫e(x2) (x dx) = ∫e^u (1/2) du

Step 4: Evaluate the integral We can now pull the constant (1/2) out of the integral:

(1/2) ∫e^u du

The integral of e^u with respect to u is simply e^u:

(1/2) ∫e^u du = (1/2) e^u + C

Here, C is the constant of integration, which is always included when evaluating indefinite integrals.

Step 5: Substitute back The final step is to substitute back for u in terms of x. We recall that u = x^2, so:

(1/2) e^u + C = (1/2) e(x2) + C

Thus, the solution to the integral ∫xe(x2) dx is (1/2) e(x2) + C. This step completes the process of u-substitution, providing us with the antiderivative of the original function. The constant of integration, C, accounts for the fact that there are infinitely many antiderivatives that differ by a constant. This result is a fundamental example in calculus and showcases the elegance and effectiveness of the u-substitution technique. By breaking down the problem into smaller, manageable steps, we have successfully evaluated the integral. In the next section, we will discuss the rationale behind this method and why it works so effectively.

Rationale Behind u-Substitution

The technique of u-substitution is a powerful tool for evaluating integrals, especially those involving composite functions. The rationale behind u-substitution lies in its ability to simplify complex integrals by reversing the chain rule of differentiation. Understanding why this method works is crucial for mastering integration techniques and applying them effectively.

The Chain Rule in Differentiation To appreciate the essence of u-substitution, let's revisit the chain rule. The chain rule states that if we have a composite function y = f(g(x)), then its derivative with respect to x is:

dy/dx = f'(g(x)) * g'(x)

In simpler terms, the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. This rule is fundamental in differential calculus and provides the basis for u-substitution in integral calculus.

Reversing the Chain Rule U-substitution can be thought of as the reverse process of the chain rule. When we have an integral of the form ∫f'(g(x)) * g'(x) dx, we can recognize that this looks like the result of applying the chain rule. The goal of u-substitution is to undo this chain rule application and find the original function f(g(x)).

The Substitution Process

  1. Choosing u: The first step in u-substitution is to choose a suitable substitution. Typically, we look for a function g(x) within the integrand whose derivative g'(x) is also present (possibly up to a constant factor). This is because we want to simplify the integral by reducing its complexity.
  2. Finding du: Once we have chosen u = g(x), we find its derivative with respect to x, du/dx = g'(x). Then, we express du in terms of dx: du = g'(x) dx.
  3. Adjusting the integral: We rewrite the original integral in terms of u. This often involves algebraic manipulation to match the du term with the remaining parts of the integrand. If necessary, we multiply or divide by constants to ensure the substitution is accurate.
  4. Evaluating the integral: After the substitution, the integral should be simpler and easier to evaluate. We integrate with respect to u, obtaining a function F(u) + C, where F is the antiderivative of the transformed integrand and C is the constant of integration.
  5. Substituting back: Finally, we substitute back for u in terms of x, replacing u with g(x) to express the result in terms of the original variable. This gives us the antiderivative in terms of x: F(g(x)) + C.

Why u-Substitution Works for ∫xe(x2) dx In the case of ∫xe(x2) dx, the integrand has the form of a composite function where e(x2) is the outer function and x^2 is the inner function. The derivative of x^2 is 2x, which is present in the integrand (up to a constant factor). This makes u-substitution an ideal method.

By substituting u = x^2, we transform the integral into a simpler form: ∫(1/2)e^u du. The factor of (1/2) arises from adjusting the integral to match du = 2x dx. The resulting integral is straightforward to evaluate, as the antiderivative of e^u is simply e^u.

In summary, u-substitution works by reversing the chain rule. It allows us to simplify integrals involving composite functions by identifying a suitable substitution that makes the integral more manageable. The key is to recognize the structure of the integrand and choose a substitution that aligns with the derivative of the inner function. This technique is a cornerstone of integral calculus and is essential for solving a wide range of problems.

Common Mistakes to Avoid

When applying u-substitution, there are several common mistakes that students and even experienced practitioners can make. Avoiding these pitfalls is crucial for ensuring accurate and efficient problem-solving. Let's explore some of the most frequent errors and how to steer clear of them.

1. Incorrect Choice of Substitution One of the most critical steps in u-substitution is choosing the right substitution. A poor choice can lead to a more complicated integral or even make the integral unsolvable using this method. The most common mistake is selecting a function u whose derivative is not present (or cannot be made present) in the integrand.

How to Avoid: Always look for a composite function within the integrand, where the derivative of the inner function is also present (up to a constant factor). In the example of ∫xe(x2) dx, choosing u = x^2 works because its derivative, 2x, is present. If you were to choose u = e(x2), the derivative would involve an additional e(x2) term, which is not as straightforward to handle.

2. Forgetting to Substitute Back A frequent error is solving the integral in terms of u but forgetting to substitute back to express the final answer in terms of the original variable, x. This is particularly important for indefinite integrals where the result should be a function of x plus a constant of integration.

How to Avoid: After evaluating the integral in terms of u, always remember to substitute back using the original substitution u = g(x). This ensures that your final answer is in the correct variable and context. For example, after finding the antiderivative (1/2)e^u + C, remember to replace u with x^2 to get (1/2)e(x2) + C.

3. Neglecting the Constant of Integration For indefinite integrals, the constant of integration, C, is a crucial part of the solution. Forgetting to include it means missing an infinite family of antiderivatives that satisfy the integral.

How to Avoid: Always add the constant of integration, C, when evaluating indefinite integrals. This constant represents the fact that the derivative of a constant is zero, so any constant could be added to an antiderivative without changing its derivative.

4. Incorrectly Adjusting the Integral When substituting du, it’s essential to correctly adjust the integral to match the differential. This often involves multiplying or dividing by constants to ensure the substitution is accurate. A common mistake is not accounting for these adjustments properly.

How to Avoid: After finding du = g'(x) dx, compare this with the terms in the original integral. If there is a constant discrepancy, adjust the integral by multiplying or dividing both sides of the equation by the appropriate constant. In our example, du = 2x dx, so we needed to divide by 2 to get (1/2) du = x dx, which matched the x dx term in the integral.

5. Misunderstanding the Limits of Integration (for Definite Integrals) When dealing with definite integrals, you must also adjust the limits of integration to correspond to the new variable u. A common mistake is using the original limits of integration without converting them to u-values.

How to Avoid: If you are evaluating a definite integral using u-substitution, either substitute back to the original variable before evaluating at the limits, or change the limits of integration to u-values. To change the limits, substitute the original limits into the substitution equation u = g(x) to find the new limits.

In conclusion, avoiding these common mistakes requires careful attention to detail and a solid understanding of the u-substitution process. By choosing the correct substitution, remembering to substitute back, including the constant of integration, adjusting the integral accurately, and handling the limits of integration properly, you can effectively solve a wide range of integrals using this powerful technique. Regular practice and mindful application of these guidelines will help you master u-substitution and tackle complex integration problems with confidence.

Practice Problems

To solidify your understanding of u-substitution and its application to integrals involving exponential functions, working through practice problems is essential. Practice problems not only reinforce the concepts but also help you develop the intuition needed to choose the appropriate substitution and avoid common mistakes. Here are a few practice problems similar to ∫xe(x2) dx, along with some guidance on how to approach them.

Problem 1: ∫x^2 e(x3) dx

Approach: In this integral, notice that the exponent of e is x^3, and the integrand also contains x^2. Recall that the derivative of x^3 is 3x^2. This suggests that u-substitution is a suitable technique. Let u = x^3, and find du. Adjust the integral accordingly, evaluate the new integral in terms of u, and then substitute back to express the final answer in terms of x.

Problem 2: ∫(e^x)/(1 + e^(2x)) dx

Approach: This integral might appear complex at first glance, but u-substitution can simplify it significantly. Notice that e^(2x) can be written as (ex)2. If we let u = e^x, then du = e^x dx. Substitute these into the integral and see how it simplifies. The resulting integral should be a standard form that you can easily evaluate.

Problem 3: ∫cos(x) e^(sin(x)) dx

Approach: This problem combines trigonometric functions with exponential functions. The key here is to recognize that the derivative of sin(x) is cos(x). Let u = sin(x), and find du. The integral should then transform into a straightforward exponential integral in terms of u. Remember to substitute back after evaluating.

Problem 4: ∫(x)/(e(x2)) dx

Approach: This integral is similar to the example we solved in the article, but with the exponential term in the denominator. Rewrite the integral as ∫x e(-x2) dx. Now, the structure is clearer. Let u = -x^2, and proceed with the u-substitution method. Pay attention to the sign when substituting and adjusting the integral.

Problem 5: ∫(e^(√x))/(√x) dx

Approach: This integral involves a square root in the exponent and the denominator. A good choice for u is √x, since its derivative will introduce a term that can cancel the √x in the denominator. Let u = √x, find du, and adjust the integral. The resulting integral should be much simpler to evaluate.

Tips for Solving Practice Problems

  1. Identify the Composite Function: Look for a function within another function. This is often a good indicator for u-substitution.
  2. Find the Derivative: Check if the derivative of the inner function is present in the integral (up to a constant factor).
  3. Choose the Substitution: Let u be the inner function, and find du.
  4. Adjust the Integral: Make sure to adjust the integral to match du, including any constant factors.
  5. Evaluate the Integral: Solve the integral in terms of u.
  6. Substitute Back: Replace u with its original expression in terms of x.
  7. Include the Constant of Integration: Don't forget to add C for indefinite integrals.

By working through these practice problems, you will gain confidence in applying u-substitution and develop a stronger understanding of integral calculus. Remember, practice is key to mastering any mathematical technique. So, grab a pencil and paper, and start integrating!

Conclusion

In this comprehensive guide, we have explored the solution to the integral ∫xe(x2) dx, a fundamental example in integral calculus that demonstrates the power of u-substitution. The journey through this integral has highlighted several crucial aspects of integration techniques, from recognizing composite functions to meticulously applying the steps of u-substitution.

We began by introducing the problem and underscoring its relevance in various scientific and engineering fields. Understanding such integrals is essential for anyone delving into calculus, as they often appear in more complex problems and applications. By breaking down the problem statement, we set the stage for a detailed solution.

Next, we presented the solution using u-substitution, a method particularly effective for integrals involving composite functions and their derivatives. The step-by-step approach—choosing the substitution, finding du, adjusting the integral, evaluating the integral, and substituting back—provided a clear roadmap for solving the problem. Each step was carefully explained to ensure a thorough understanding of the process.

Delving deeper, we discussed the rationale behind u-substitution. This technique essentially reverses the chain rule of differentiation, allowing us to simplify complex integrals. Understanding this underlying principle is vital for effectively applying u-substitution and recognizing when it is the appropriate method to use.

We also addressed common mistakes to avoid when using u-substitution. Errors such as incorrect choice of substitution, forgetting to substitute back, neglecting the constant of integration, and incorrectly adjusting the integral can lead to incorrect solutions. By being aware of these pitfalls, you can improve your accuracy and problem-solving skills.

Finally, we provided practice problems to help solidify your understanding. These problems offer an opportunity to apply the techniques learned and develop the intuition needed to tackle similar integrals. Practice is a cornerstone of mastering any mathematical concept, and integral calculus is no exception.

In summary, the integral ∫xe(x2) dx serves as an excellent illustration of the power and elegance of u-substitution. By mastering this technique, you will be well-equipped to handle a wide range of integrals involving exponential and composite functions. The key takeaways from this guide include the importance of recognizing the structure of the integrand, choosing an appropriate substitution, carefully applying the steps of u-substitution, and practicing regularly to build confidence and proficiency. As you continue your journey in calculus, remember that a solid understanding of integration techniques like u-substitution will be invaluable in solving more advanced problems and exploring the vast applications of calculus in various fields.