antiderivative of 1 x 3

Guri Bee

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21-10-2024

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Understanding the Antiderivative of 1/x^3

The expression "1/x^3" can be rewritten as x^-3. Finding its antiderivative involves reversing the process of differentiation.

What is an Antiderivative?

In calculus, the antiderivative of a function is another function whose derivative is the original function. It's like finding the "opposite" of a derivative.

Finding the Antiderivative of x^-3

1. Power Rule: The power rule for antiderivatives states:

   • The antiderivative of x^n is (x^(n+1))/(n+1) + C, where n is any real number except -1 and C is an arbitrary constant of integration.

2. Applying the Power Rule:

   • Applying the power rule to x^-3, we get: (x^(-3+1))/(-3+1) + C = x^-2 / -2 + C = -1/(2x^2) + C

Therefore, the antiderivative of 1/x^3 is -1/(2x^2) + C, where C is an arbitrary constant.

Why the Constant of Integration?

The constant of integration, C, arises because the derivative of any constant is zero. This means that any function of the form -1/(2x^2) + C will have the same derivative.

Example:

Let's say we have the function f(x) = -1/(2x^2) + 5.

• The derivative of f(x) is f'(x) = 1/x^3.
• Similarly, the function g(x) = -1/(2x^2) - 2 also has the same derivative.

Practical Applications:

Understanding antiderivatives is crucial in various fields:

• Physics: Finding velocity and displacement from acceleration.
• Engineering: Calculating areas under curves and volumes of solids.
• Economics: Determining marginal cost and revenue.

Note:

This explanation is based on information found on the GitHub repository [link to relevant repository]. However, this article goes beyond the basic information by providing examples and explaining the significance of the constant of integration.