Integral Of X X 1 1 2

Solving the Integral of x/(x² + 1)²: A Step-by-Step Guide

This article will guide you through the process of solving the definite integral ∫ x/(x² + 1)² dx. This integral requires a specific integration technique, making it a valuable example for calculus students. We'll break down the solution step-by-step, explaining the reasoning behind each step and highlighting key concepts. Understanding this process will enhance your skills in solving similar complex integrals.

Understanding the Problem:

The integral ∫ x/(x² + 1)² dx presents a challenge because it's not a straightforward power rule integration. The presence of the (x² + 1)² in the denominator requires a strategic approach. We'll employ a substitution method to simplify the integrand.

Step 1: Substitution

The most effective approach here is u-substitution. Let's define:

• u = x² + 1

Now, we need to find the derivative of u with respect to x:

• du/dx = 2x

Solving for dx, we get:

• dx = du/(2x)

Step 2: Substituting into the Integral

Substitute 'u' and 'dx' into the original integral:

∫ x/(x² + 1)² dx = ∫ x/u² * (du/(2x))

Notice that the 'x' in the numerator cancels out with the 'x' in the denominator:

= ∫ 1/(2u²) du

Step 3: Simplifying and Integrating

The integral is now much simpler:

= (1/2) ∫ u⁻² du

Applying the power rule of integration (∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C):

= (1/2) * (u⁻¹)/(-1) + C

= -1/(2u) + C

Step 4: Back-Substitution

Remember that u = x² + 1. Substitute this back into the equation:

= -1/(2(x² + 1)) + C

Step 5: The Final Solution

Therefore, the indefinite integral of x/(x² + 1)² is:

-1/(2(x² + 1)) + C

where 'C' is the constant of integration.

Key Concepts and Further Exploration:

• U-Substitution: This is a crucial technique for simplifying integrals. Mastering u-substitution is essential for tackling more complex integration problems.
• Power Rule of Integration: Understanding the power rule is fundamental to solving a wide range of integrals.
• Definite Integrals: To solve a definite integral (with upper and lower limits), you would evaluate the antiderivative at those limits and subtract the results. The constant of integration ('C') cancels out in definite integrals.
• Trigonometric Substitution: For integrals involving expressions like √(a² - x²), √(a² + x²), or √(x² - a²), trigonometric substitution is a powerful method.
• Partial Fraction Decomposition: For integrals with rational functions (ratios of polynomials), partial fraction decomposition can simplify the integrand.

This detailed explanation provides a comprehensive understanding of how to solve the integral x/(x² + 1)². By practicing similar problems and exploring the related concepts mentioned above, you can significantly improve your calculus skills. Remember to always carefully choose the appropriate integration technique based on the structure of the integrand.