Anti Derivative Of Square Root Of X

Finding the Antiderivative of the Square Root of x: A Comprehensive Guide

Finding the antiderivative of the square root of x, or ∫√x dx, is a fundamental concept in calculus. This guide will walk you through the process step-by-step, explaining the underlying principles and providing practical examples. Understanding this seemingly simple integral is crucial for mastering more complex integration techniques later on.

What is an Antiderivative?

Before diving into the specifics, let's clarify what an antiderivative is. Simply put, it's the reverse of a derivative. If you differentiate a function, you find its derivative; if you find the antiderivative, you find the original function (plus a constant of integration). This process is also known as indefinite integration.

Solving ∫√x dx

The square root of x can be rewritten as x^1/2. This allows us to apply the power rule of integration, which states:

∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C

where 'n' is any real number except -1, and 'C' is the constant of integration (crucial because the derivative of a constant is zero).

Applying the power rule to our problem:

1. Rewrite the function: ∫√x dx = ∫x^1/2 dx

2. Apply the power rule: (x^(1/2)+1)/((1/2)+1) + C

3. Simplify: (x^3/2)/(3/2) + C

4. Further simplification: (2/3)x^3/2 + C

Therefore, the antiderivative of √x is (2/3)x^3/2 + C.

Understanding the Constant of Integration (C)

The constant of integration, 'C', is a crucial component of the antiderivative. This is because the derivative of any constant is zero. Therefore, infinitely many functions could have the same derivative. The constant 'C' accounts for this ambiguity. For example, (2/3)x^3/2 + 5, (2/3)x^3/2 - 2, and (2/3)x^3/2 + 100 all have the same derivative: √x.

Practical Applications and Extensions

Understanding the antiderivative of √x forms the foundation for solving more complex integrals. For instance, consider integrals involving terms like 2√x, or even more complicated expressions containing the square root of x. By applying techniques like u-substitution and integration by parts (topics for another discussion!), you can tackle these more advanced problems.

Example Problem:

Let's find the definite integral of √x from 1 to 4: ∫₁⁴ √x dx

1. Find the antiderivative: We already know this is (2/3)x^3/2 + C

2. Evaluate at the limits of integration: [(2/3)(4)^3/2 + C] - [(2/3)(1)^3/2 + C]

3. Simplify: [(2/3)(8) + C] - [(2/3)(1) + C] = 16/3 - 2/3 = 14/3

Therefore, the definite integral of √x from 1 to 4 is 14/3.

Conclusion:

Finding the antiderivative of √x is a straightforward yet crucial step in mastering calculus. By understanding the power rule and the significance of the constant of integration, you gain a strong foundation for tackling more complex integration problems. Remember to practice regularly to solidify your understanding and build confidence in your problem-solving abilities.