Evaluating ∫x³(x⁴-2)³⁸ Dx With U-Substitution A Step-by-Step Guide

In calculus, evaluating integrals is a fundamental skill, and one powerful technique for tackling complex integrals is u-substitution. This method allows us to simplify the integrand by introducing a new variable, making the integration process more manageable. This article will discuss the step-by-step solution to evaluate the integral ∫x³(x⁴-2)³⁸ dx by using the substitution method, where u = x⁴ - 2. We will see how this substitution simplifies the integral and allows us to find the antiderivative effectively. This approach is a cornerstone of integral calculus, enabling us to solve a wide array of problems involving complex functions. Let's dive into the solution and explore the elegance of u-substitution in action.

Understanding U-Substitution

Before we dive into the specific integral, let's briefly discuss the u-substitution technique. U-substitution, also known as substitution or change of variables, is a method used to simplify integrals by replacing a complex expression within the integrand with a single variable, u. The key idea is to identify a function and its derivative within the integral, which allows for a simplification that makes the integration process easier. This method is particularly useful when dealing with composite functions, where one function is nested inside another. By substituting u for the inner function, we can often transform the integral into a more recognizable form that can be easily integrated using standard integration rules. For example, if we have an integral of the form ∫f(g(x))g'(x) dx, we can let u = g(x), then du = g'(x) dx, and the integral becomes ∫f(u) du, which is often simpler to evaluate. This technique is a cornerstone of integral calculus, enabling us to solve a wide array of problems involving complex functions. Mastering u-substitution is crucial for anyone studying calculus, as it is frequently used in various applications and is a building block for more advanced integration techniques. Through careful selection of the substitution variable, we can transform seemingly intractable integrals into manageable ones, unlocking the path to their solutions. The power of u-substitution lies in its ability to simplify the structure of the integrand, making the task of finding the antiderivative significantly easier. It is a testament to the elegance and efficiency of mathematical problem-solving techniques.

Step-by-Step Solution

To evaluate the integral ∫x³(x⁴-2)³⁸ dx, we'll follow a structured approach using the u-substitution method. This will involve several key steps, each designed to simplify the integral until we can easily find its antiderivative. First, we identify the appropriate substitution, then we find the differential of our new variable. Next, we rewrite the original integral in terms of our new variable, which should result in a simpler form. We then evaluate this simplified integral, and finally, we substitute back to get our answer in terms of the original variable. This process, while methodical, is incredibly powerful, allowing us to tackle integrals that would otherwise be extremely difficult to solve. By carefully following each step, we can transform a complex-looking integral into a straightforward one, highlighting the elegance and efficiency of u-substitution. The methodical nature of this approach not only helps in solving the specific problem at hand but also provides a template for tackling similar integrals in the future. The ability to break down a complex problem into manageable steps is a crucial skill in mathematics, and u-substitution is an excellent example of this principle in action. The transformation from the original integral to the simplified form is often the most insightful part of the process, revealing the underlying structure of the integrand and paving the way for a clean and elegant solution. Let's go through each of these steps in detail.

1. Identify the Substitution

In the given integral, ∫x³(x⁴-2)³⁸ dx, we can observe that the expression (x⁴-2) is raised to a power. This suggests that we can make a substitution to simplify the expression. Let's set u = x⁴ - 2. This is a crucial first step, as choosing the right substitution is key to simplifying the integral. The goal is to select a substitution that will transform the integral into a more manageable form, ideally one that we can solve using standard integration rules. In this case, the expression inside the parentheses, (x⁴-2), seems like a good candidate because its derivative will involve x³, which also appears in the integral. This observation is a typical indicator that u-substitution will be effective. The ability to recognize such patterns is a skill that develops with practice and a deep understanding of the underlying principles of calculus. The choice of u = x⁴ - 2 not only simplifies the integrand but also sets the stage for the subsequent steps, where we will find du and rewrite the integral in terms of u. This strategic selection of the substitution variable is a hallmark of effective problem-solving in calculus.

2. Find du

Now that we have chosen our substitution, u = x⁴ - 2, we need to find the differential du. To do this, we differentiate u with respect to x: du/dx = d/dx (x⁴ - 2). Applying the power rule, we get du/dx = 4x³. Next, we solve for du by multiplying both sides by dx: du = 4x³ dx. This step is critical because it allows us to relate the differential of the new variable, du, to the differential of the original variable, dx. The ability to manipulate differentials is a fundamental skill in calculus and is essential for performing substitutions correctly. The equation du = 4x³ dx provides the bridge between the original integral and the integral in terms of u. It tells us how the infinitesimal changes in u are related to the infinitesimal changes in x. This relationship is what allows us to rewrite the integral in a different form, making it easier to solve. The calculation of du is a straightforward application of differentiation rules, but its role in the substitution process is profound. It is the key that unlocks the simplification of the integral.

3. Rewrite the Integral

With du = 4x³ dx, we need to rewrite the original integral ∫x³(x⁴-2)³⁸ dx in terms of u. Notice that we have x³ dx in the integral, but our expression for du contains 4x³ dx. To match the terms, we can divide both sides of the du equation by 4: (1/4)du = x³ dx. Now we can substitute u and (1/4)du into the integral. Replacing (x⁴-2) with u and x³ dx with (1/4)du, we get: ∫(u)³⁸ (1/4)du. This can be further simplified by moving the constant (1/4) outside the integral: (1/4)∫u³⁸ du. This transformation is the heart of the u-substitution method. By carefully substituting u and du, we have converted a complex integral involving a power of a binomial into a much simpler integral involving a power of a single variable. This simplified integral is now in a form that we can easily evaluate using the power rule for integration. The process of rewriting the integral highlights the power of substitution in simplifying mathematical problems. By changing the variables, we have unveiled the underlying structure of the integral, making it accessible to standard integration techniques. The transformation from the original integral to the (1/4)∫u³⁸ du is a testament to the elegance and efficiency of u-substitution.

4. Evaluate the Integral in Terms of u

Now we need to evaluate the simplified integral, (1/4)∫u³⁸ du. We can use the power rule for integration, which states that ∫xⁿ dx = (x^(n+1))/(n+1) + C, where C is the constant of integration. Applying this rule to our integral, we get: (1/4) * (u^(38+1))/(38+1) + C = (1/4) * (u³⁹/39) + C. This can be further simplified to u³⁹/156 + C. This step is a direct application of the power rule, a fundamental concept in integral calculus. The power rule allows us to find the antiderivative of a power function, which is a common task in integration. The constant of integration, C, is crucial because it represents the family of functions that have the same derivative. Without the constant of integration, we would only be finding one particular antiderivative, rather than the general solution. The evaluation of the integral in terms of u is a straightforward calculation, but it is a key step in the overall process. It transforms the integral from a symbolic expression into a function, representing the antiderivative of the original integrand. The simplicity of the result in terms of u underscores the effectiveness of the u-substitution method. It demonstrates how a carefully chosen substitution can transform a complex integral into a simple one, making the task of finding the antiderivative significantly easier.

5. Substitute Back

The final step is to substitute back the original expression for u in terms of x. Recall that we defined u = x⁴ - 2. So, we replace u with (x⁴ - 2) in our result: (x⁴ - 2)³⁹/156 + C. This gives us the final answer in terms of the original variable, x. Therefore, the integral ∫x³(x⁴-2)³⁸ dx evaluates to (x⁴ - 2)³⁹/156 + C. This step is crucial because it returns the solution to the original problem. While we simplified the integral by introducing a new variable, the ultimate goal is to express the antiderivative in terms of the original variable. The substitution back is a straightforward replacement, but it completes the solution process. It transforms the antiderivative in terms of u into the antiderivative in terms of x. The final result, (x⁴ - 2)³⁹/156 + C, represents the general solution to the integral. It is a function of x whose derivative is the original integrand, x³(x⁴-2)³⁸. The constant of integration, C, reminds us that there are infinitely many functions that satisfy this condition, differing only by a constant. The successful completion of this step marks the end of the integration process, demonstrating the power and elegance of the u-substitution method.

Conclusion

In conclusion, we have successfully evaluated the integral ∫x³(x⁴-2)³⁸ dx using the u-substitution method. By setting u = x⁴ - 2, we simplified the integral and found the antiderivative to be (x⁴ - 2)³⁹/156 + C. This example demonstrates the power and elegance of u-substitution in simplifying complex integrals. U-substitution is a fundamental technique in integral calculus, and mastering it is essential for solving a wide variety of integration problems. It allows us to transform seemingly intractable integrals into manageable ones by carefully selecting a substitution that simplifies the integrand. The process involves identifying a suitable substitution, finding the differential of the new variable, rewriting the integral in terms of the new variable, evaluating the simplified integral, and finally, substituting back to express the result in terms of the original variable. This methodical approach, while requiring practice and understanding of calculus principles, is a powerful tool in the arsenal of any mathematician or scientist. The ability to apply u-substitution effectively opens the door to solving more complex problems in calculus and related fields. The elegance of this method lies in its ability to reveal the underlying structure of an integral, making it accessible to standard integration techniques. The solution we obtained, (x⁴ - 2)³⁹/156 + C, represents the general antiderivative of the given integrand, highlighting the importance of including the constant of integration. The successful application of u-substitution in this case serves as a testament to its utility and importance in the field of integral calculus.