Mastering the Power Rule for Integration | A Comprehensive Guide to Finding Antiderivatives of Functions Raised to a Power

Power Rule for Integration

The Power Rule for Integration is a formula used to find the antiderivative of a function raised to a power

The Power Rule for Integration is a formula used to find the antiderivative of a function raised to a power. It is the reverse process of differentiation.

The Power Rule for Integration states that if we have a function of the form f(x) = x^n, where n is any real number other than -1, then the antiderivative of f(x) can be found by adding one to the power and dividing by the new power.

Mathematically, if n is not equal to -1, then the integral of x^n dx is (x^(n+1))/(n+1) + C, where C is the constant of integration.

Let’s consider some examples to apply the Power Rule for Integration:

Example 1: Find the antiderivative of f(x) = 3x^2.

Using the Power Rule for Integration, we add one to the power and divide by the new power. In this case, n = 2, so the antiderivative is (3/3)x^(2+1) + C = x^3 + C.

Example 2: Find the antiderivative of f(x) = 5x^(-4).

Again, using the Power Rule for Integration, we add one to the power and divide by the new power. In this case, n = -4, so the antiderivative is (5/-4+1)x^(-4+1) + C = -(5/3)x^(-3) + C.

Example 3: Find the antiderivative of f(x) = x^0.

Here, n = 0, but the Power Rule for Integration does not hold for n = -1. When n = 0, the antiderivative becomes (1/0+1)x^0+1 + C = x + C.

It is important to note that the Power Rule for Integration only works for powers other than -1. When the power is -1, we need to use a different method called logarithmic differentiation to find the antiderivative.

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