Solving The Integral Of 1/(3x+1)^15 A Step-by-Step Guide

Integrating functions, especially those involving powers of linear expressions, is a fundamental skill in calculus. This article provides a comprehensive guide on how to solve the integral of 1/(3x+1)^15, a classic example that demonstrates the power of u-substitution. We will break down the process step-by-step, ensuring clarity and understanding for students and enthusiasts alike. This detailed exploration not only covers the specific solution but also reinforces the underlying principles of integration techniques. Grasping these techniques is crucial for tackling more complex integrals and applications in various fields of mathematics, physics, and engineering. So, let's dive into the intricacies of integration and master this essential skill.

Understanding the Integral

Before diving into the solution, let's understand the integral we aim to solve: ∫ dx / (3x + 1)^15. This integral represents the antiderivative of the function f(x) = 1 / (3x + 1)^15. Our primary goal is to find a function F(x) such that its derivative F'(x) equals f(x). Recognizing the structure of the integrand is crucial. We have a function of the form 1 / (ax + b)^n, where a = 3, b = 1, and n = 15. This form strongly suggests that u-substitution is the most effective method for solving this integral. U-substitution simplifies the integral by replacing a complex expression with a single variable, making the integration process more manageable. Understanding the underlying principles and recognizing these patterns are essential for tackling various integration problems. By mastering these techniques, you'll gain a solid foundation in calculus and be well-prepared for more advanced mathematical concepts and applications.

The U-Substitution Method

The u-substitution method is a powerful technique for simplifying integrals. It involves substituting a part of the integrand with a new variable, 'u,' and adjusting the differential accordingly. In our case, the expression (3x + 1) inside the power function is a good candidate for substitution. Let's set u = 3x + 1. This substitution simplifies the denominator considerably. The next step is to find the differential du in terms of dx. Differentiating both sides of u = 3x + 1 with respect to x, we get du/dx = 3. Solving for dx, we find dx = du / 3. Now we have both u and dx in terms of the original variable x and the new variable u. We can substitute these expressions back into the original integral, transforming it into a simpler form that is easier to integrate. This method is based on the chain rule in differentiation, effectively reversing the process to find the antiderivative. Mastering u-substitution is crucial for handling a wide variety of integrals, and it forms the basis for more advanced integration techniques.

Performing the Substitution

Now that we have defined our substitution, u = 3x + 1 and dx = du / 3, let's substitute these into the original integral: ∫ dx / (3x + 1)^15. Replacing (3x + 1) with u and dx with du / 3, the integral becomes ∫ (du / 3) / u^15. We can rewrite this as (1/3) ∫ du / u^15. This constant factor of 1/3 can be moved outside the integral, simplifying the expression further. The integral now looks like (1/3) ∫ u^(-15) du. This form is much simpler to handle, as it involves a power of u. We've successfully transformed the original integral into a more manageable form using u-substitution. The next step involves applying the power rule for integration, which we will discuss in detail in the following section. This transformation highlights the power of u-substitution in simplifying complex integrals, making them solvable using basic integration rules.

Applying the Power Rule for Integration

The power rule for integration is a fundamental rule that states: ∫x^n dx = (x^(n+1)) / (n+1) + C, where n ≠ -1 and C is the constant of integration. In our transformed integral, (1/3) ∫ u^(-15) du, we have a power of u, specifically u^(-15). Applying the power rule, we add 1 to the exponent and divide by the new exponent: ∫ u^(-15) du = u^(-15 + 1) / (-15 + 1) + C = u^(-14) / (-14) + C. Now, we can substitute this result back into our expression: (1/3) ∫ u^(-15) du = (1/3) * [u^(-14) / (-14) + C]. This step involves a straightforward application of the power rule, demonstrating its versatility in handling integrals of power functions. The constant of integration, C, is crucial because it represents the family of antiderivatives. The derivative of any constant is zero, so adding a constant does not change the derivative of the antiderivative. Therefore, we include C to account for all possible antiderivatives. This step is a critical part of the integration process, ensuring a complete and accurate solution.

Substituting Back and Simplifying

After applying the power rule, we have (1/3) * [u^(-14) / (-14) + C]. Now, we need to substitute back our original expression for u, which was u = 3x + 1. Replacing u with (3x + 1), we get (1/3) * [(3x + 1)^(-14) / (-14) + C]. Next, we can simplify this expression. Multiplying the constants, we get -(1/42) * (3x + 1)^(-14) + (1/3)C. Since (1/3)C is still an arbitrary constant, we can simply replace it with a new constant, C'. Thus, the expression becomes -(1/42) * (3x + 1)^(-14) + C'. We can further rewrite (3x + 1)^(-14) as 1 / (3x + 1)^14. So, the final simplified form of the integral is -1 / [42(3x + 1)^14] + C'. This step demonstrates the importance of substituting back and simplifying the expression to obtain the final antiderivative in terms of the original variable, x. The simplification process ensures that the answer is presented in the most concise and understandable form, making it easier to use in further calculations or applications. This complete process highlights the step-by-step approach to solving integrals using u-substitution and the power rule.

The Final Solution

After performing the u-substitution, applying the power rule, substituting back, and simplifying, we arrive at the final solution for the integral ∫ dx / (3x + 1)^15. The solution is: -1 / [42(3x + 1)^14] + C, where C is the constant of integration. This result represents the antiderivative of the function 1 / (3x + 1)^15. It is crucial to include the constant of integration, C, because there are infinitely many antiderivatives that differ by a constant. This solution demonstrates the effectiveness of the u-substitution method in simplifying complex integrals. By breaking down the problem into manageable steps, we were able to apply the power rule and arrive at the final answer. This process not only provides the solution to this specific integral but also reinforces the general principles of integration techniques. Understanding these techniques is essential for tackling a wide range of calculus problems and applications in various fields.

Conclusion

In conclusion, we have successfully solved the integral ∫ dx / (3x + 1)^15 using the u-substitution method and the power rule for integration. This detailed step-by-step guide has demonstrated the process of identifying a suitable substitution, transforming the integral, applying the power rule, substituting back, and simplifying the result. The final solution, -1 / [42(3x + 1)^14] + C, represents the antiderivative of the given function. This example highlights the importance of mastering fundamental integration techniques for solving more complex problems in calculus. By understanding and applying these principles, students and enthusiasts can gain a deeper appreciation for the power and elegance of calculus. Integration is a cornerstone of mathematical analysis, with applications spanning various fields, including physics, engineering, economics, and computer science. Therefore, a solid understanding of integration techniques is invaluable for anyone pursuing careers in these disciplines. This comprehensive guide serves as a valuable resource for anyone seeking to enhance their understanding of integration and problem-solving skills in calculus.