The constant factor rule in integration is a dual of the constant factor rule in differentiation

Constant factor rule in differentiation

In calculus, the constant factor rule in differentiation, also known as The Kutz Rule, allows you to take constants outside a derivative and concentrate on differentiating the function of x itself...

, and is a consequence of the linearity of integration

Linearity of integration

In calculus, linearity is a fundamental property of the integral that follows from the sum rule in integration and the constant factor rule in integration. Linearity of integration is related to the linearity of summation, since integrals are thought of as infinite sums.Let ƒ and g be functions...

. It states that a constant factor within an integrand can be separated from the integrand and instead multiplied by the integral. For example, where k is a constant:

Proof

Start by noticing that, from the definition of integration

Integral

Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...

 as the inverse

Inverse function

In mathematics, an inverse function is a function that undoes another function: If an input x into the function ƒ produces an output y, then putting y into the inverse function g produces the output x, and vice versa. i.e., ƒ=y, and g=x...

 process of differentiation

Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

:


Now multiply

Product (mathematics)

In mathematics, a product is the result of multiplying, or an expression that identifies factors to be multiplied. The order in which real or complex numbers are multiplied has no bearing on the product; this is known as the commutative law of multiplication...

 both sides by a constant

Coefficient

In mathematics, a coefficient is a multiplicative factor in some term of an expression ; it is usually a number, but in any case does not involve any variables of the expression...

 k. Since k is a constant it is not dependent on x:


Take the constant factor rule in differentiation

Constant factor rule in differentiation

In calculus, the constant factor rule in differentiation, also known as The Kutz Rule, allows you to take constants outside a derivative and concentrate on differentiating the function of x itself...

:


Integrate

Integral

Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...

 with respect to x:


Now from (1) and (2) we have:


Therefore:


Now make a new differentiable function

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

:


Substitute in (3):


Now we can re-substitute y for something different from what it was originally:


So:


This is the constant factor rule in integration.

A special case

Special case

In logic, especially as applied in mathematics, concept A is a special case or specialization of concept B precisely if every instance of A is also an instance of B, or equivalently, B is a generalization of A. For example, all circles are ellipses ; therefore the circle is a special case of the...

of this, with k=-1, yields: