Integral Of The Square Root Of X

Solving the Integral of the Square Root of x: A Step-by-Step Guide

This article provides a comprehensive guide on how to solve the indefinite integral of the square root of x, a fundamental concept in calculus. We'll break down the process step-by-step, explaining the underlying principles and offering helpful tips for similar integration problems. Understanding this integral is crucial for various applications in mathematics, physics, and engineering.

Understanding the Problem:

We aim to find the indefinite integral of √x, which can be written mathematically as:

∫√x dx

This represents the antiderivative of the function f(x) = √x. Remember, integration is the reverse process of differentiation.

The Power Rule of Integration:

The key to solving this integral lies in the power rule of integration. The power rule states that:

∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C

where 'n' is any real number except -1, and 'C' is the constant of integration.

Applying the Power Rule:

First, rewrite √x as x¹⁄². Now, we can apply the power rule:

1. Rewrite the integrand: Our integral becomes ∫x¹⁄² dx

2. Apply the power rule: Using the formula above, with n = 1/2, we get:

   (x¹⁄²⁺¹)/(1/2 + 1) + C

3. Simplify: Adding the exponents gives us:

   (x³/²) / (3/2) + C

4. Further simplification: To simplify further, we can multiply the numerator and denominator by 2/3:

   (2/3)x³/² + C

Therefore, the indefinite integral of √x is (2/3)x³/² + C

Understanding the Constant of Integration (C):

The constant of integration, 'C', is crucial. Since the derivative of a constant is zero, any constant added to the antiderivative will still result in the original function when differentiated. This means there's a family of functions that all have the same derivative, differing only by the constant.

Example Application:

Let's say we need to find the area under the curve y = √x from x = 1 to x = 4. We would use the definite integral:

∫₁⁴ √x dx = [(2/3)x³/²]₁⁴ = (2/3)(4)³/² - (2/3)(1)³/² = (16/3) - (2/3) = 14/3

This calculation shows a practical application of our integral.

Extending the Concept:

This foundational understanding can be extended to more complex integrals involving the square root of x, such as those including other functions within the integrand. Mastering the basic principles will equip you to tackle more challenging problems in integration. Remember to always rewrite radical expressions using fractional exponents before applying the power rule.

Conclusion:

Successfully solving the integral of the square root of x hinges on correctly applying the power rule of integration and understanding the significance of the constant of integration. This seemingly simple integral serves as a building block for more advanced integration techniques. Practice applying this method to solidify your understanding and prepare for more complex calculus problems.