What is the standard method to find the integral of a constant multiplied by a function?

The correct approach to finding the integral of a constant multiplied by a function is to first integrate the function and then multiply the result by that constant. This is a direct application of the properties of integrals, specifically the constant multiple rule. According to this rule, if you have a constant ( k ) and a function ( f(x) ), the integral of ( k \cdot f(x) ) over an interval can be expressed as:

[

\int k \cdot f(x) , dx = k \cdot \int f(x) , dx

]

This implies that you can factor the constant out of the integral, allowing for simpler computation.

Understanding this property is crucial, especially in calculus, because it simplifies the process when dealing with polynomials, trigonometric functions, and many other types of functions.

Other methods presented do not align with this foundational rule. For example, simply adding the constant to the integral of the function would not yield the correct answer, as it neglects the primary influence that the constant has on the scale of the integral. Dividing the function by the constant before integrating fundamentally changes the function being integrated, thus altering the outcome.

Therefore, multiplying the integral of