1. Find The Indefinite Integral Of The Function Y = 6x² + 3/x. 2. What Is The Indefinite Integral Of Xe^(x²)? 3. Determine The Indefinite Integral Of (sin√x)/√x.

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In the realm of calculus, integration stands as a fundamental operation, the inverse of differentiation. It's the process of finding the area under a curve, accumulating quantities, and solving a myriad of problems in physics, engineering, economics, and beyond. This article serves as a comprehensive guide to mastering integrals, walking you through various techniques and strategies for tackling complex problems. We'll delve into specific examples, providing step-by-step solutions and explanations to solidify your understanding. Whether you're a student grappling with calculus concepts or a professional seeking a refresher, this guide will equip you with the tools and knowledge to confidently navigate the world of integrals. Let's embark on this journey together and unravel the intricacies of integration.

1. Evaluating the Integral: y = 6x² + 3/x

Understanding the Problem

In this section, we'll focus on evaluating the integral of the function y = 6x² + 3/x. This problem combines polynomial and rational terms, requiring a solid grasp of basic integration rules and properties. To effectively tackle this integral, we'll break it down into simpler parts, applying the power rule and the natural logarithm rule. Our goal is to find the antiderivative of the given function, which represents the family of functions whose derivative is equal to 6x² + 3/x. This process involves reversing the rules of differentiation, a skill that is crucial for solving a wide range of calculus problems. By the end of this section, you'll have a clear understanding of how to integrate such expressions and the importance of including the constant of integration.

Step-by-Step Solution

To evaluate the integral of y = 6x² + 3/x, we'll follow these steps:

  1. Rewrite the function: First, we can rewrite the function as y = 6x² + 3x⁻¹. This form makes it easier to apply the power rule of integration.
  2. Apply the power rule and constant multiple rule: The integral of 6x² is found by increasing the exponent by 1 and dividing by the new exponent, resulting in (6x³)/3. Similarly, the integral of 3x⁻¹ requires a different approach as the power rule doesn't apply directly when the exponent is -1. Instead, we recognize that the integral of x⁻¹ is ln|x|.
  3. Integrate each term separately: Applying these rules, the integral of 6x² is 2x³, and the integral of 3x⁻¹ is 3ln|x|.
  4. Add the constant of integration: Since we're finding the indefinite integral, we must add a constant of integration, C. This constant represents the family of functions that have the same derivative.
  5. Final Result: Combining these results, the integral of y = 6x² + 3/x is 2x³ + 3ln|x| + C.

Key Concepts Applied

  • Power Rule: The power rule of integration states that ∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C, where n ≠ -1. This rule is fundamental for integrating polynomial terms.
  • Constant Multiple Rule: The constant multiple rule allows us to pull out a constant from the integral, i.e., ∫kf(x) dx = k∫f(x) dx, where k is a constant. This simplifies the integration process.
  • Integral of 1/x: The integral of 1/x is ln|x| + C. This is a crucial rule for integrating rational functions where the denominator is x.
  • Constant of Integration: The constant of integration, C, is added to the indefinite integral because the derivative of a constant is zero. It represents the family of functions with the same derivative.

2. Evaluating the Integral: ∫xe^(x²) dx

Understanding the Problem

Next, let's tackle the integral of ∫xe^(x²) dx. This integral involves a composite function, where the exponential function e is raised to the power of x². Such integrals often require the technique of u-substitution, a powerful method for simplifying integrals by changing the variable of integration. The key to u-substitution is identifying a suitable substitution that simplifies the integrand, making it easier to integrate. In this case, we'll see how choosing u = x² transforms the integral into a more manageable form. This section will provide a detailed walkthrough of the u-substitution process, highlighting the steps involved and the reasoning behind them. By mastering this technique, you'll be able to solve a wide variety of integrals involving composite functions.

Step-by-Step Solution

To evaluate the integral ∫xe^(x²) dx, we'll use u-substitution:

  1. Choose a substitution: Let u = x². This substitution is strategic because the derivative of x² is 2x, which appears in the integral (up to a constant).
  2. Find du: Differentiating u with respect to x, we get du/dx = 2x, which implies du = 2x dx.
  3. Adjust the integral: We need to express the integral in terms of u. Notice that our integral has x dx, but our du is 2x dx. To match, we can rewrite du as (1/2) du = x dx.
  4. Substitute: Now, substitute u = x² and (1/2) du = x dx into the integral: ∫e^u (1/2) du.
  5. Integrate: The integral of (1/2)e^u with respect to u is (1/2)e^u + C.
  6. Substitute back: Replace u with x² to get the final answer: (1/2)e^(x²) + C.

Key Concepts Applied

  • U-Substitution: U-substitution is a technique used to simplify integrals by substituting a function and its derivative. The goal is to transform the integral into a simpler form that can be easily integrated.
  • Composite Functions: Composite functions are functions within functions, like e^(x²). U-substitution is particularly useful for integrals involving composite functions.
  • Changing the Variable of Integration: U-substitution involves changing the variable of integration from x to u, which requires finding the relationship between dx and du.
  • Substituting Back: After integrating with respect to u, it's essential to substitute back the original variable (x in this case) to express the result in terms of the original variable.

3. Finding the Integral: ∫(sin√x)/√x dx

Understanding the Problem

In this final section, we'll delve into finding the integral of ∫(sin√x)/√x dx. This integral presents a trigonometric function within a composite function, making it another excellent candidate for u-substitution. Similar to the previous example, identifying the appropriate substitution is crucial. Here, the square root function inside the sine function suggests a potential substitution. We'll walk through the process of choosing the substitution, finding the differential, and transforming the integral. This example will further solidify your understanding of u-substitution and its application to a variety of integral problems. By the end of this section, you'll be well-equipped to recognize and solve integrals involving trigonometric and composite functions.

Step-by-Step Solution

To find the integral ∫(sin√x)/√x dx, we'll again employ u-substitution:

  1. Choose a substitution: Let u = √x. This substitution simplifies the sine function's argument.
  2. Find du: Differentiating u with respect to x, we get du/dx = 1/(2√x), which implies du = (1/(2√x)) dx.
  3. Adjust the integral: We need to match the integral with our du. The integral has (1/√x) dx, and our du has (1/(2√x)) dx. To match, we can rewrite du as 2 du = (1/√x) dx.
  4. Substitute: Substitute u = √x and 2 du = (1/√x) dx into the integral: ∫sin(u) * 2 du.
  5. Integrate: The integral of 2sin(u) with respect to u is -2cos(u) + C.
  6. Substitute back: Replace u with √x to get the final answer: -2cos(√x) + C.

Key Concepts Applied

  • Strategic Substitution: Choosing the right substitution is key to simplifying integrals. Look for functions and their derivatives within the integral.
  • Trigonometric Integrals: Integrals involving trigonometric functions often require specific substitutions or trigonometric identities to solve.
  • Differential Adjustment: Adjusting the differential (du) to match the integral is a crucial step in u-substitution.
  • Final Solution: Remember to substitute back the original variable to express the solution in terms of the original problem.

Mastering integrals is a cornerstone of calculus and a vital skill for numerous applications. In this comprehensive guide, we've explored various techniques and strategies for solving complex integral problems. We've dissected examples involving polynomial, rational, exponential, and trigonometric functions, emphasizing the power of u-substitution in simplifying composite functions. Each step-by-step solution has been designed to illuminate the underlying concepts and build your confidence in tackling a wide range of integrals. Remember, practice is paramount. By working through numerous examples and applying these techniques, you'll develop the intuition and expertise to excel in the world of integration. As you continue your journey in calculus, keep these principles in mind, and you'll find that the complexities of integration become increasingly manageable and even enjoyable.