Integral Of X Arcsin X

Integrating x arcsin x: A Comprehensive Guide

Finding the integral of x arcsin x might seem daunting at first glance, but with a systematic approach and a good understanding of integration techniques, it becomes manageable. This comprehensive guide will walk you through the process, explaining each step clearly and providing a deep dive into the underlying mathematical principles. We'll cover the solution method, delve into the relevant properties of inverse trigonometric functions, and address frequently asked questions. This detailed explanation will equip you with the skills to tackle similar integration problems.

Understanding the Problem: ∫x arcsin x dx

Our goal is to evaluate the definite integral of x multiplied by the inverse sine function of x, denoted as ∫x arcsin x dx. This integral requires the application of integration by parts, a powerful technique for integrating products of functions.

Integration by Parts: The Key Technique

Integration by parts is based on the product rule of differentiation. The formula is:

∫u dv = uv - ∫v du

Choosing the right 'u' and 'dv' is crucial for successful application. The general strategy is to choose 'u' as a function that simplifies when differentiated and 'dv' as a function that is easily integrated.

Step-by-Step Solution: Finding ∫x arcsin x dx

1. Choosing u and dv:

   Let's set:

   • u = arcsin x => du = 1/(√(1-x²)) dx
   • dv = x dx => v = (1/2)x²

2. Applying the Integration by Parts Formula:

   Substituting these values into the integration by parts formula, we get:

   ∫x arcsin x dx = (1/2)x² arcsin x - ∫(1/2)x² * (1/(√(1-x²))) dx

3. Simplifying and Solving the Remaining Integral:

   The remaining integral, ∫(1/2)x²/(√(1-x²)) dx, is more challenging. We'll use a trigonometric substitution to solve it.

4. Trigonometric Substitution:

   Let's substitute x = sin θ. This implies dx = cos θ dθ. The integral then transforms into:

   ∫(1/2)sin²θ/(√(1-sin²θ)) * cos θ dθ = (1/2)∫sin²θ dθ

5. Solving the Trigonometric Integral:

   We can use the power-reducing formula for sin²θ:

   sin²θ = (1 - cos(2θ))/2

   Substituting this into the integral gives:

   (1/2)∫(1 - cos(2θ))/2 dθ = (1/4)∫(1 - cos(2θ)) dθ

   Integrating term by term, we obtain:

   (1/4)[θ - (1/2)sin(2θ)] + C

6. Back-Substitution:

   Remember that x = sin θ, so θ = arcsin x. Also, sin(2θ) = 2sinθcosθ = 2x√(1-x²). Substituting back into the expression, we get:

   (1/4)[arcsin x - x√(1-x²)] + C

7. Combining the Results:

   Finally, combine this result with the first part of the integration by parts equation:

   ∫x arcsin x dx = (1/2)x² arcsin x - (1/4)[arcsin x - x√(1-x²)] + C

   This simplifies to:

   ∫x arcsin x dx = (1/2)x² arcsin x + (1/4)x√(1-x²) - (1/4)arcsin x + C

Explanation of Steps and Underlying Principles

• Integration by Parts: This is a fundamental technique for integrating products of functions. The success hinges on the strategic choice of 'u' and 'dv'. Choosing 'u' as arcsin x allows us to simplify the derivative, while choosing 'dv' as x dx leads to a simple integral for 'v'.

• Trigonometric Substitution: This substitution is a classic technique for handling integrals involving expressions of the form √(a² - x²) or similar. The substitution x = sin θ transforms the integral into a trigonometric integral which is often easier to solve.

• Power-Reducing Formula: This formula is a useful tool for simplifying integrals involving powers of trigonometric functions. It allows us to express sin²θ in terms of a simpler trigonometric function, making the integration more straightforward.

• Back-Substitution: This is the crucial step where we revert back to the original variable, x, after solving the integral in terms of θ. It's important to correctly express trigonometric functions of 2θ in terms of x.

Further Considerations and Extensions

This solution provides a definitive answer to the integral. However, depending on the context, further simplification might be needed or alternative approaches could be explored. For example, you could analyze the definite integral within specific limits.

Frequently Asked Questions (FAQ)

• Q: Why is the choice of 'u' and 'dv' so important in integration by parts?

  A: The choice of 'u' and 'dv' directly impacts the complexity of the resulting integral. A poor choice can lead to a more complicated integral than the original one. The goal is to choose 'u' such that its derivative is simpler and 'dv' such that its integral is easily obtained.

• Q: Are there other ways to solve this integral?

  A: While integration by parts with trigonometric substitution is a common and effective method, other advanced techniques might be applicable in specific cases. However, this method provides a clear and relatively straightforward path to the solution.

• Q: What if the integral had different limits?

  A: If the integral has defined limits (a definite integral), you would substitute those limits into the final antiderivative obtained above and subtract the results to find the numerical value of the definite integral.

• Q: How can I improve my skills in solving similar integration problems?

  A: Practice is key! Work through a variety of integration problems, focusing on mastering different integration techniques such as integration by parts, trigonometric substitution, and partial fraction decomposition. Also, understand the underlying principles and properties of the functions involved.

Conclusion

Integrating x arcsin x successfully requires a combination of integration techniques including integration by parts and trigonometric substitution. By carefully selecting the appropriate 'u' and 'dv' in the integration by parts step and applying the correct trigonometric substitution and power-reducing formulas, we can arrive at the solution. Understanding these techniques and practicing their application will strengthen your calculus skills and enable you to tackle more complex integration problems confidently. Remember to always double-check your work and ensure that your solution is consistent with the properties of the functions involved. Mastering integration is a journey of practice and understanding; with persistent effort, you will undoubtedly improve your skills.