Why should we add a constant c to the result of integration?

Question:

Why should we add a constant c to the result of integration?

Indefinite Integral:

Indefinite integral is employed to get the set of primitive functions of a particular function.

The primitive functions of a function {eq}f(x) {/eq} are those functions resulting in the function {eq}f(x) {/eq} when their derivatives are taken.

Adding a constant {eq}C {/eq} in an indefinite integral is a standard procedure in taking antiderivatives.

Answer and Explanation:

The reason we add a constant {eq}C {/eq} after integrating a function {eq}f(x) {/eq} is to represent the family of functions whose derivatives are the function {eq}f(x) {/eq}.

Note that when we derive a constant we get {eq}0 {/eq}, so whatever constant we add to the resulting antiderivative will give us a function whose derivative is the function being integrated.

Consider the function {eq}f(x) = x {/eq}.

Its indefinite integral is {eq}F(x) = \displaystyle \frac{x^2}{2} {/eq} and the derivative of {eq}F(x) = \displaystyle \frac{x^2}{2} {/eq} is {eq}f(x) = x {/eq}, so {eq}F(x) = \displaystyle \frac{x^2}{2} {/eq} is an antiderivative of {eq}f(x) {/eq}.

But we can have another antiderivative simply by adding any constant.

Take for example {eq}F(x) = \displaystyle \frac{x^2}{2}+3 {/eq}.

Its derivative is also {eq}f(x) = x {/eq} as the derivative of the constant {eq}3 {/eq}, or any constant {eq}C {/eq}, is {eq}0 {/eq}.


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Indefinite Integrals as Anti Derivatives

from Math 104: Calculus

Chapter 12 / Lesson 11
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