Evaluating The Integral Of (1+x)^2 / √x Step-by-Step

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Introduction

In this article, we will delve into the step-by-step evaluation of the integral: ∫ [(1+x)^2 / √x] dx. This integral, while appearing complex at first glance, can be solved using algebraic manipulation and basic integration techniques. Understanding the fundamentals of calculus is crucial for tackling such problems, and we'll break down each step to ensure clarity. Our approach will involve expanding the numerator, simplifying the expression, and then applying the power rule for integration. This methodical approach will not only help in solving this specific integral but also in building a strong foundation for handling more complex integrals in the future. The goal is to provide a comprehensive guide that readers can follow, regardless of their prior experience with calculus, making the process of integration more accessible and less intimidating.

Rewriting the Integral

To begin, we need to simplify the integrand, which is the function inside the integral. Our integrand is (1+x)^2 / √x. The first step is to expand the numerator, (1+x)^2. Using the binomial formula or simply multiplying (1+x) by itself, we get (1+x)^2 = 1 + 2x + x^2. Now, we can rewrite the integral as ∫ [(1 + 2x + x^2) / √x] dx. The next step is to divide each term in the numerator by √x, which is equivalent to x^(1/2). This gives us ∫ [1/√x + 2x/√x + x^2/√x] dx. To simplify further, we can express each term with fractional exponents: 1/√x = x^(-1/2), 2x/√x = 2x^(1 - 1/2) = 2x^(1/2), and x^2/√x = x^(2 - 1/2) = x^(3/2). Thus, the integral becomes ∫ [x^(-1/2) + 2x^(1/2) + x^(3/2)] dx. This form is much easier to integrate term by term, as each term is now a simple power of x. Mastering the art of algebraic manipulation is key to simplifying complex integrals, making them more manageable and allowing us to apply standard integration rules effectively.

Applying the Power Rule

Now that we have simplified the integral to ∫ [x^(-1/2) + 2x^(1/2) + x^(3/2)] dx, we can apply the power rule for integration. The power rule states that ∫ x^n dx = (x^(n+1)) / (n+1) + C, where n is any real number except -1, and C is the constant of integration. Let's apply this rule to each term in our integral. For the first term, x^(-1/2), we have n = -1/2. Thus, ∫ x^(-1/2) dx = (x^(-1/2 + 1)) / (-1/2 + 1) + C = (x^(1/2)) / (1/2) + C = 2x^(1/2) + C. For the second term, 2x^(1/2), we have n = 1/2. So, ∫ 2x^(1/2) dx = 2 * (x^(1/2 + 1)) / (1/2 + 1) + C = 2 * (x^(3/2)) / (3/2) + C = (4/3)x^(3/2) + C. Finally, for the third term, x^(3/2), we have n = 3/2. Therefore, ∫ x^(3/2) dx = (x^(3/2 + 1)) / (3/2 + 1) + C = (x^(5/2)) / (5/2) + C = (2/5)x^(5/2) + C. By applying the power rule to each term, we have successfully integrated each part of the expression. The power rule is a cornerstone of integral calculus, and understanding its application is essential for solving a wide range of integrals.

Combining the Results

Having integrated each term separately, we can now combine the results to obtain the final solution for the integral ∫ [x^(-1/2) + 2x^(1/2) + x^(3/2)] dx. We found that ∫ x^(-1/2) dx = 2x^(1/2) + C, ∫ 2x^(1/2) dx = (4/3)x^(3/2) + C, and ∫ x^(3/2) dx = (2/5)x^(5/2) + C. Adding these together, we get 2x^(1/2) + (4/3)x^(3/2) + (2/5)x^(5/2) + C, where C represents the constant of integration. This constant is crucial because the derivative of a constant is zero, meaning there are infinitely many functions that could have this derivative. We can also rewrite the result in terms of square roots to make it more visually appealing and easier to understand. Recall that x^(1/2) = √x, x^(3/2) = x√x, and x^(5/2) = x^2√x. Substituting these back into our result, we get 2√x + (4/3)x√x + (2/5)x^2√x + C. This final expression represents the antiderivative of the original integrand. Combining individual results carefully and remembering the constant of integration are key steps in completing the integration process. This final form not only gives us the general solution but also provides a clearer understanding of the function's behavior.

Final Solution

In summary, the integral ∫ [(1+x)^2 / √x] dx evaluates to 2√x + (4/3)x√x + (2/5)x^2√x + C, where C is the constant of integration. This solution was obtained by first expanding the numerator, simplifying the integrand by dividing each term by √x, and then applying the power rule for integration to each term. We then combined the results and expressed the final answer in a simplified form, both with fractional exponents and in terms of square roots. The process of evaluating integrals often involves a combination of algebraic manipulation, application of integration rules, and careful simplification. By mastering these techniques, one can confidently tackle a wide range of integration problems. This exercise demonstrates how breaking down a complex problem into smaller, manageable steps can lead to a clear and accurate solution. Understanding each step, from the initial simplification to the final combination of results, is essential for building a strong foundation in calculus. This final solution not only answers the specific problem but also reinforces the broader principles of integration, providing a valuable learning experience for anyone studying calculus.