antiderivative x 3

2 min read 28-02-2025

The antiderivative, also known as the indefinite integral, is a fundamental concept in calculus. It's the reverse process of differentiation. This article will guide you through finding the antiderivative of x³, explaining the process and providing examples. Understanding antiderivatives is crucial for solving various problems in mathematics, science, and engineering.

Understanding Antiderivatives

Before diving into the specifics of x³, let's briefly review the core concept. The antiderivative of a function f(x) is a function F(x) whose derivative is f(x), i.e., F'(x) = f(x). Importantly, the antiderivative isn't unique; adding a constant (C) to any antiderivative still yields an antiderivative. This constant represents the family of functions that share the same derivative.

The Power Rule for Antiderivatives

The power rule is the key to finding the antiderivative of x³. The power rule for integration states that the antiderivative of xⁿ (where n ≠ -1) is (xⁿ⁺¹)/(n+1) + C, where C is the constant of integration.

Calculating the Antiderivative of x³

Applying the power rule to x³ (where n=3):

Identify the exponent: In x³, the exponent (n) is 3.
Apply the power rule: Add 1 to the exponent (3+1=4) and divide the term by the new exponent: (x⁴)/4.
Add the constant of integration: Remember to always include the constant of integration, C.

Therefore, the antiderivative of x³ is (x⁴)/4 + C.

Examples and Applications

Let's look at a few examples to solidify our understanding:

Example 1: Find the antiderivative of 5x³.

Following the power rule, we get: (5x⁴)/4 + C. The constant 5 remains as a coefficient.

Example 2: Solve the indefinite integral ∫x³ dx.

The integral symbol (∫) is another way to denote finding the antiderivative. Therefore, ∫x³ dx = (x⁴)/4 + C.

Example 3: Real-World Application

Imagine you're working with a physics problem involving acceleration. If the acceleration a(t) = 3t², the velocity v(t) is found by taking the antiderivative: v(t) = ∫3t² dt = t³ + C. The constant C represents the initial velocity.

Common Mistakes to Avoid

Forgetting the constant of integration (C): This is crucial because the derivative of a constant is zero. Omitting C leads to an incomplete answer.
Incorrectly applying the power rule: Double-check your exponent addition and division. A common error is forgetting to divide by the new exponent.
Confusing derivatives and antiderivatives: Remember they are inverse operations.

Conclusion

Finding the antiderivative of x³ is a straightforward application of the power rule. By mastering this fundamental concept, you’ll be well-equipped to tackle more complex integration problems. Remember the power rule, include the constant of integration, and practice regularly to build your confidence and understanding. The ability to find antiderivatives is a cornerstone skill in calculus and numerous related fields.