antiderivative of x 2

2 min read 28-02-2025

The antiderivative, also known as the indefinite integral, of a function is a family of functions whose derivative is the original function. Finding the antiderivative is a fundamental concept in calculus. This article will guide you through finding the antiderivative of x², explaining the process and providing examples.

Understanding Antiderivatives

Before diving into the specific case of x², let's review the core idea. If F'(x) = f(x), then F(x) is an antiderivative of f(x). It's important to note that there's not just one antiderivative; any constant added to F(x) will also have the same derivative. This is why we represent the antiderivative with a general constant "+C".

The Power Rule for Antiderivatives

The power rule is a crucial tool for finding antiderivatives of functions in the form xⁿ. The rule states:

∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C (where n ≠ -1)

This formula will be our key to solving the antiderivative of x².

Finding the Antiderivative of x²

Now, let's apply the power rule to find the antiderivative of x². In this case, n = 2. Substituting into the power rule formula:

∫x² dx = (x²⁺¹)/(2+1) + C

Simplifying:

∫x² dx = (x³)/3 + C

Therefore, the antiderivative of x² is (x³/3) + C, where C represents the arbitrary constant of integration. This constant accounts for the infinite number of possible antiderivatives.

Examples and Applications

Let's explore a few examples to solidify our understanding:

Example 1: Find the antiderivative of 5x².

Using the constant multiple rule (∫cf(x)dx = c∫f(x)dx), we have:

∫5x² dx = 5∫x² dx = 5[(x³)/3] + C = (5x³)/3 + C

Example 2: Find the antiderivative of x² + 2x.

Here, we use the sum/difference rule (∫[f(x) ± g(x)]dx = ∫f(x)dx ± ∫g(x)dx):

∫(x² + 2x) dx = ∫x² dx + ∫2x dx = (x³/3) + x² + C

Applications: Antiderivatives are fundamental to many areas of calculus and beyond. They're crucial for:

Calculating areas under curves: Using definite integrals (which involve evaluating antiderivatives at specific limits).
Solving differential equations: These equations involve derivatives, and finding solutions often necessitates finding antiderivatives.
Physics: Finding displacement from velocity, or velocity from acceleration.

Definite Integrals and the Antiderivative of x²

While we've focused on indefinite integrals (finding the general antiderivative), the concept extends to definite integrals. A definite integral calculates the area under a curve between two specified limits. For x², this would be:

∫_a^b x² dx = [(x³)/3]^b_a = (b³/3) - (a³/3)

This provides a numerical value representing the area.

Conclusion: Mastering the Antiderivative of x²

The antiderivative of x² is a foundational concept in calculus. Understanding the power rule and its application, as demonstrated in this article, allows you to find antiderivatives and solve various problems across different fields. Remember the crucial "+C" representing the constant of integration, which accounts for the infinite family of antiderivatives for a given function. Practice applying these concepts to various problems to solidify your understanding.