Understanding Integration | Antiderivative of xⁿ and the Power Rule

∫ xⁿ dx

The integral of x to the power of n, written as ∫ xⁿ dx, represents the antiderivative or the indefinite integral of the function xⁿ with respect to x

The integral of x to the power of n, written as ∫ xⁿ dx, represents the antiderivative or the indefinite integral of the function xⁿ with respect to x.

To find the integral of xⁿ, we can follow a simple formula:

∫ xⁿ dx = (x^(n+1))/(n+1) + C

In this formula, C represents the constant of integration, which accounts for the fact that an indefinite integral has an infinite number of solutions, differing by a constant value.

Let’s go through an example to illustrate this:

Example: Find the integral of x² dx.

In this case, n = 2, so we’ll use the formula:

∫ x² dx = (x^(2+1))/(2+1) + C

Simplifying the expression, we get:

∫ x² dx = (x³)/3 + C

So, the antiderivative of x² is (x³)/3 + C.

It’s important to note that the power rule for integration, which we used here, is applicable for all real values of n except for n = -1. For n = -1, the integral becomes the natural logarithm: ∫(1/x) dx = ln|x| + C.

If you have any other specific questions or need further explanations, feel free to ask!

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