antiderivative of x/3

2 min read 25-02-2025

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 calculating the antiderivative of x/3.

Understanding Antiderivatives

Before diving into the specific problem, let's review the core concept. If F'(x) = f(x), then F(x) is an antiderivative of f(x). Note that the antiderivative is not unique; adding a constant (C) to any antiderivative still results in an antiderivative. This is because the derivative of a constant is always zero.

Calculating the Antiderivative of x/3

The function we're working with is f(x) = x/3. To find its antiderivative, we'll use the power rule of integration. The power rule states that the antiderivative of xⁿ is (xⁿ⁺¹)/(n+1) + C, where n ≠ -1 and C is the constant of integration.

Let's rewrite x/3 as (1/3)x¹. Now, we can apply the power rule:

Identify n: In our case, n = 1.
Apply the power rule: The antiderivative is [(1/3)x⁽¹⁺¹⁾] / (1+1) + C.
Simplify: This simplifies to [(1/3)x²] / 2 + C = (1/6)x² + C.

Therefore, the antiderivative of x/3 is (1/6)x² + C.

Verifying the Result

We can verify our answer by differentiating the result. If we take the derivative of (1/6)x² + C, we should get back our original function, x/3.

Using the power rule of differentiation, the derivative of (1/6)x² is (1/6) * 2x = x/3. The derivative of the constant C is 0. Thus, our calculation is correct.

Practical Applications

Understanding antiderivatives is crucial in various fields, including:

Physics: Calculating displacement from velocity or velocity from acceleration.
Engineering: Determining the area under a curve (integration).
Economics: Modeling marginal cost and revenue functions.

Common Mistakes to Avoid

Forgetting the constant of integration (C): This is a common error. Remember that the antiderivative represents a family of functions, and the constant reflects this.
Incorrect application of the power rule: Double-check your exponent calculations.
Misunderstanding the concept: Make sure you understand that the antiderivative is the inverse operation of differentiation.

Conclusion

Finding the antiderivative of x/3 is a straightforward application of the power rule. The result is (1/6)x² + C, where C is the constant of integration. Remember to always verify your answer by differentiating and to be mindful of common mistakes. Mastering this fundamental concept is key to further progress in calculus and its applications.