Evaluating ∫(2x + 1)(x² + X) Dx A Step-by-Step Guide

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This article provides a detailed walkthrough of evaluating the integral ∫(2x + 1)(x² + x) dx. We will explore various methods, including u-substitution, to arrive at the solution. This is a common type of problem encountered in introductory calculus courses, and understanding the techniques involved is crucial for mastering integration. Whether you're a student grappling with calculus concepts or simply seeking to refresh your integration skills, this guide offers a clear and comprehensive explanation.

Understanding the Integral: ∫(2x + 1)(x² + x) dx

When faced with the integral ∫(2x + 1)(x² + x) dx, the initial approach might involve expanding the product and then integrating term by term. While this method is valid, it can be more cumbersome than necessary. A more elegant and efficient solution lies in recognizing the relationship between the factors within the integral. Specifically, notice that the derivative of x² + x is 2x + 1. This observation strongly suggests the application of u-substitution, a powerful technique for simplifying integrals.

Before diving into the solution, let's briefly discuss the fundamental concepts at play. Integration, at its core, is the reverse process of differentiation. When we evaluate an integral, we seek a function whose derivative is the integrand (the expression being integrated). The power rule of integration, a cornerstone of calculus, states that ∫xⁿ dx = (x^(n+1))/(n+1) + C, where n ≠ -1 and C is the constant of integration. This rule will be instrumental in our solution. Moreover, the constant of integration, C, is a critical component of any indefinite integral. It represents the family of functions that differ only by a constant term and whose derivatives all equal the integrand. Neglecting the constant of integration leads to an incomplete solution.

The method of u-substitution is based on the chain rule of differentiation. It allows us to simplify integrals by making a strategic substitution that transforms the integrand into a more manageable form. In this case, we will let u = x² + x. This substitution not only simplifies the expression but also elegantly incorporates the (2x + 1) term. The key to u-substitution lies in correctly identifying a suitable 'u' and its derivative, 'du'. When the integrand contains both a function and its derivative (or a constant multiple of its derivative), u-substitution is often the optimal approach.

Method 1: U-Substitution

This method is generally the most efficient way to solve this integral. The key is recognizing that the derivative of x² + x is present (or a constant multiple of it) within the integrand. Let's break down the steps:

  1. Identify 'u': Let u = x² + x. This is a crucial step, and choosing the right 'u' can significantly simplify the integration process. In this case, we've selected x² + x because its derivative appears in the integral.
  2. Find 'du': Differentiate both sides of the equation u = x² + x with respect to x. This gives us du/dx = 2x + 1. Multiplying both sides by dx, we get du = (2x + 1) dx. Notice how this expression perfectly matches the remaining part of the integrand, making our substitution seamless.
  3. Substitute: Replace x² + x with u and (2x + 1) dx with du in the original integral. This transforms the integral into ∫u du. This substitution is the heart of the method, converting a complex integral into a simple one.
  4. Integrate: The integral ∫u du is straightforward. Using the power rule, we find the antiderivative to be (u²)/2 + C, where C is the constant of integration. Remember, the constant of integration is essential for representing the family of antiderivatives.
  5. Substitute Back: Replace 'u' with its original expression, x² + x. This gives us the final result: ((x² + x)²)/2 + C. This step is crucial for expressing the solution in terms of the original variable, x.

Therefore, ∫(2x + 1)(x² + x) dx = ((x² + x)²)/2 + C.

Method 2: Expanding and Integrating Term by Term

Alternatively, we can solve the integral by expanding the product first and then integrating each term separately. While this method works, it's generally more involved than u-substitution in this particular case. However, it's a valuable technique to understand for other integrals.

  1. Expand the product: Multiply (2x + 1) and (x² + x): (2x + 1)(x² + x) = 2x³ + 2x² + x² + x = 2x³ + 3x² + x. This expansion transforms the integrand into a polynomial, making term-by-term integration possible.
  2. Integrate term by term: Integrate each term of the expanded polynomial: ∫(2x³ + 3x² + x) dx = ∫2x³ dx + ∫3x² dx + ∫x dx. This step applies the linearity of integration, allowing us to handle each term independently.
  3. Apply the power rule: Use the power rule of integration (∫xⁿ dx = (x^(n+1))/(n+1) + C) for each term:
    • ∫2x³ dx = 2 * (x⁴/4) = (x⁴/2)
    • ∫3x² dx = 3 * (x³/3) = x³
    • ∫x dx = (x²/2)
  4. Combine the results: Sum the results of each integration and add the constant of integration: (x⁴/2) + x³ + (x²/2) + C. This gives us the antiderivative in a slightly different form than the u-substitution method.
  5. Verify Equivalence: While the result (x⁴/2) + x³ + (x²/2) + C looks different from ((x² + x)²)/2 + C, they are mathematically equivalent. Expanding ((x² + x)²)/2 gives us (x⁴ + 2x³ + x²)/2 = (x⁴/2) + x³ + (x²/2). This demonstrates that both methods lead to the same family of antiderivatives, just expressed in different forms. The constant of integration, C, accounts for the potential vertical shift between these seemingly different functions.

Therefore, ∫(2x + 1)(x² + x) dx = (x⁴/2) + x³ + (x²/2) + C.

Comparing the Methods

Both u-substitution and expanding and integrating term by term are valid methods for evaluating the integral ∫(2x + 1)(x² + x) dx. However, u-substitution proves to be more efficient in this specific case. It streamlines the process by directly addressing the relationship between the factors in the integrand. By recognizing that (2x + 1) is the derivative of (x² + x), u-substitution allows us to transform the integral into a simpler form (∫u du) with minimal algebraic manipulation.

Expanding and integrating term by term, while correct, involves more steps. The initial expansion of the product leads to a polynomial with multiple terms, each requiring separate integration. This increases the chance of making algebraic errors. Furthermore, while the final result is equivalent, it may not be immediately obvious without further algebraic manipulation to show the equivalence. For more complex integrals, this difference in efficiency can become even more pronounced.

In general, when faced with an integral, it's beneficial to first consider u-substitution if you notice a function and its derivative (or a constant multiple thereof) within the integrand. If u-substitution doesn't seem applicable or if the integrand is a simple polynomial, expanding and integrating term by term might be a suitable alternative. The ability to recognize the optimal method is a key skill in integral calculus.

Key Takeaways

  • The integral ∫(2x + 1)(x² + x) dx can be solved using u-substitution or by expanding and integrating term by term.
  • U-substitution is generally the more efficient method for this type of integral because it leverages the relationship between the factors in the integrand.
  • Expanding and integrating term by term is a valid alternative but involves more algebraic steps.
  • The constant of integration, C, is a crucial component of indefinite integrals and represents a family of antiderivatives.
  • Mastering u-substitution is essential for solving a wide range of integrals in calculus.
  • Understanding the relationship between differentiation and integration is fundamental to calculus. Integration is the inverse operation of differentiation, allowing us to find functions whose derivatives are known.
  • The power rule of integration (∫xⁿ dx = (x^(n+1))/(n+1) + C) is a cornerstone of integral calculus and is frequently used in conjunction with other integration techniques.
  • Practice is key to mastering integration techniques. The more problems you solve, the better you will become at recognizing patterns and choosing the most efficient method.

Conclusion

Evaluating the integral ∫(2x + 1)(x² + x) dx provides a valuable illustration of the power and elegance of integration techniques, particularly u-substitution. By understanding the underlying principles and practicing various methods, you can confidently tackle a wide range of integration problems. Remember to always consider the constant of integration and to choose the most efficient method for each specific integral. This comprehensive guide has hopefully provided you with the necessary tools and insights to approach similar problems with confidence and clarity. Remember to always double-check your work and consider alternative methods to ensure the accuracy of your solution. The world of integral calculus is vast and fascinating, and with practice and dedication, you can master its intricacies.