Introduction:

In calculus, the power rule for integration is a fundamental technique used to find the antiderivative or integral of a function that involves a power of a variable.



Understanding the Power Rule for Integration:



The power rule is applied to functions of the form f(x) = x^n, where n is a constant and not equal to -1.

This rule simplifies the process of integrating polynomial functions.

Formula of the Power Rule for Integration:



The formula states that the integral of x^n with respect to x is: ∫x^n dx = (x^(n+1))/(n+1) + C, where:

n is the exponent in the original function.

(n+1) is the new exponent after integration.

(n+1) in the denominator is used to balance the equation.

C represents the constant of integration.

Example:



To integrate f(x) = x²:

Apply the power rule: ∫x² dx = (x^(2+1))/(2+1) + C.

Simplify: ∫x² dx = x³/3 + C.

Key Points to Remember:



The power rule for integration is not applicable when n = -1, as it leads to a logarithmic function.

Always add the constant of integration (C) when integrating indefinite integrals.