Integral Of 1 X 1 2

Understanding the Integral of 1/(x^1 * x^2)

This article explores how to solve the integral of 1/(x¹ * x²), simplifying the expression and applying fundamental integration techniques. The seemingly simple expression presents a valuable opportunity to review core calculus concepts and proper integration procedures. This guide will break down the steps clearly, clarifying any potential confusion surrounding the problem.

Simplifying the Expression:

Before tackling the integration, let's simplify the expression within the integral: 1/(x¹ * x²). Recall that x¹ is simply x. Therefore, our expression becomes:

1/(x * x²) = 1/x³ = x⁻³

Now, we have a much cleaner expression to integrate: ∫x⁻³ dx

Applying the Power Rule of Integration:

The power rule of integration states that the integral of xⁿ dx is (xⁿ⁺¹)/(n+1) + C, where n is any real number except -1, and C represents the constant of integration.

In our case, n = -3. Applying the power rule:

∫x⁻³ dx = (x⁻³⁺¹)/(-3 + 1) + C

Simplifying further:

(x⁻²)/(-2) + C = -x⁻²/2 + C

Final Result and Understanding the Constant of Integration:

Our final result for the indefinite integral of 1/(x¹ * x²) is:

-1/(2x²) + C

The constant of integration, C, is crucial. Because the derivative of a constant is always zero, adding any constant to the antiderivative will still result in the original function when differentiated. Therefore, we include 'C' to represent the family of antiderivatives.

Practical Applications and Further Considerations:

Understanding this type of integral is fundamental in various fields, including:

• Physics: Calculating work done by a variable force.
• Engineering: Solving differential equations related to fluid dynamics or electrical circuits.
• Economics: Modeling certain types of growth or decay.

This seemingly straightforward integral provides a solid exercise in applying fundamental calculus principles. Mastering these basic integration techniques is essential for tackling more complex problems later on. Remember to always simplify the expression before integrating and to never forget the constant of integration. By consistently practicing these steps, you'll build a strong foundation in integral calculus.