Calculating The Indefinite Integral Of (6x + 6) / (2x^2 + 4x + 12)

In the realm of calculus, indefinite integrals serve as a cornerstone for understanding the behavior of functions and their antiderivatives. Tackling indefinite integrals requires a blend of algebraic manipulation, pattern recognition, and a firm grasp of integration techniques. This article delves into the step-by-step process of calculating the indefinite integral of the function (6x + 6) / (2x^2 + 4x + 12), ensuring a clear and comprehensive understanding of each stage. By the end of this exploration, you'll not only have the solution to this specific problem but also gain valuable insights into tackling similar integration challenges. This is a foundational skill in various fields, including physics, engineering, and economics, where understanding rates of change and accumulation is paramount. So, let's embark on this journey of mathematical discovery and unravel the intricacies of indefinite integration. Our goal is to present a comprehensive and accessible guide that empowers you to confidently tackle indefinite integrals, regardless of their complexity. Remember, the key to mastering calculus lies in consistent practice and a deep understanding of fundamental principles. Let's dive in and unlock the secrets of indefinite integration together.

Step 1: Simplifying the Integrand

The first crucial step in tackling any integral is to simplify the integrand as much as possible. This often involves algebraic manipulations that make the integral more amenable to standard integration techniques. In our case, the integrand is (6x + 6) / (2x^2 + 4x + 12). A keen eye will spot a common factor in both the numerator and the denominator. We can factor out a 6 from the numerator, resulting in 6(x + 1). Similarly, we can factor out a 2 from the denominator, yielding 2(x^2 + 2x + 6). Now, our integral transforms into ∫ [6(x + 1)] / [2(x^2 + 2x + 6)] dx. The beauty of this simplification lies in the ability to cancel the common factor of 2 between the numerator and the denominator. This yields ∫ [3(x + 1)] / (x^2 + 2x + 6) dx. By pulling the constant 3 out of the integral, we further simplify the expression to 3 ∫ (x + 1) / (x^2 + 2x + 6) dx. This seemingly small step is a critical maneuver, making the integral significantly easier to handle in the subsequent steps. It exemplifies the importance of looking for opportunities to simplify before diving into more complex integration methods. This simplified form lays the groundwork for a more strategic approach to solving the integral. The ability to recognize and execute these simplifications is a hallmark of a proficient calculus practitioner.

Step 2: Recognizing the Potential for u-Substitution

Now that we've simplified the integrand, the next step is to identify a suitable integration technique. A powerful tool in our arsenal is u-substitution, which is particularly effective when the integrand contains a function and its derivative (or a constant multiple thereof). Examining our simplified integral, 3 ∫ (x + 1) / (x^2 + 2x + 6) dx, we notice a promising pattern. The denominator, x^2 + 2x + 6, is a quadratic expression. If we consider its derivative, we get 2x + 2, which is remarkably similar to the numerator, x + 1 (ignoring the constant multiple for now). This observation strongly suggests that u-substitution is the ideal method to tackle this integral. The beauty of u-substitution lies in its ability to transform a complex integral into a simpler one by introducing a new variable, 'u', which represents a carefully chosen part of the original integrand. In this case, our choice of 'u' will be the denominator, x^2 + 2x + 6. This strategic selection allows us to effectively 'undo' the chain rule of differentiation, making the integration process significantly more manageable. Recognizing these patterns and strategically applying u-substitution is a cornerstone of integral calculus. It showcases the importance of understanding the relationships between functions and their derivatives in the context of integration.

Step 3: Performing the u-Substitution

Having identified u-substitution as the appropriate technique, we now proceed with the actual substitution. Let's define our 'u' as u = x^2 + 2x + 6. The next step is to find the differential 'du', which is the derivative of 'u' with respect to 'x' multiplied by 'dx'. Differentiating u = x^2 + 2x + 6, we get du/dx = 2x + 2. Multiplying both sides by 'dx', we obtain du = (2x + 2) dx. Now, notice that our numerator in the integral is (x + 1) dx. We can rewrite du as du = 2(x + 1) dx. To match our numerator, we divide both sides of the equation by 2, resulting in (1/2) du = (x + 1) dx. This is a crucial step as it allows us to directly replace (x + 1) dx in the original integral with (1/2) du. Now, we can rewrite our integral 3 ∫ (x + 1) / (x^2 + 2x + 6) dx in terms of 'u'. Substituting u = x^2 + 2x + 6 and (1/2) du = (x + 1) dx, we get 3 ∫ (1/u) (1/2) du. This transformation has simplified our integral considerably. We now have a much more manageable integral in terms of the variable 'u'. The power of u-substitution lies in this very transformation – converting a complex integral into a simpler, more recognizable form. The ability to meticulously perform this substitution is a testament to a solid understanding of differential calculus and its inverse relationship with integration.

Step 4: Integrating with Respect to u

With our integral now expressed in terms of 'u', we can proceed with the integration. We have 3 ∫ (1/u) (1/2) du. We can pull the constant (1/2) out of the integral, giving us (3/2) ∫ (1/u) du. The integral of 1/u with respect to 'u' is a standard result: it's the natural logarithm of the absolute value of 'u', denoted as ln|u|. Therefore, ∫ (1/u) du = ln|u| + C, where C is the constant of integration. Applying this to our integral, we get (3/2) ln|u| + C. Remember, the constant of integration, C, is essential for indefinite integrals, as it accounts for the family of antiderivatives that differ by a constant. We have successfully integrated with respect to 'u', but our final answer needs to be expressed in terms of the original variable, 'x'. The next step involves substituting back the original expression for 'u'. This meticulous process of integrating in terms of 'u' and then back-substituting highlights the elegance and efficiency of u-substitution as an integration technique. It demonstrates the power of transforming a problem into a more manageable form before tackling the core calculation.

Step 5: Substituting Back to x and Final Answer

Having integrated with respect to 'u', our final step is to substitute back the original expression for 'u' in terms of 'x'. Recall that we defined u = x^2 + 2x + 6. Substituting this back into our result, (3/2) ln|u| + C, we get (3/2) ln|x^2 + 2x + 6| + C. Now, we need to consider the absolute value. The expression inside the absolute value, x^2 + 2x + 6, is a quadratic. To determine its sign, we can analyze its discriminant. The discriminant is given by b^2 - 4ac, where a = 1, b = 2, and c = 6. The discriminant is 2^2 - 4 * 1 * 6 = 4 - 24 = -20. Since the discriminant is negative and the coefficient of x^2 is positive, the quadratic expression x^2 + 2x + 6 is always positive. Therefore, we can remove the absolute value signs. This gives us our final answer: (3/2) ln(x^2 + 2x + 6) + C. This completes the calculation of the indefinite integral. The addition of '+ C' is crucial for indefinite integrals, as it represents the constant of integration. Omitting this would mean we're only giving one specific antiderivative and not the entire family of antiderivatives. This final step emphasizes the importance of meticulous attention to detail and a thorough understanding of the properties of functions within the context of integration. The successful completion of this substitution marks the culmination of our integration journey, providing us with the complete and accurate solution.

Final Answer:

(3/2) ln(x^2 + 2x + 6) + C