Lesson 7, Topic 2
In Progress

Constant of Integration

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If $$\scriptsize y = x^4, \therefore \normalsize \frac{dy}{dx} \scriptsize = 4x^3 \; and \; \int 4x^3 dx = x^4$$

Also $$\scriptsize y = x^4 + 2, \therefore \normalsize \frac{dy}{dx} \scriptsize = 4x^3 \; and \; \int 4x^3 dx = x^4 + 2$$

and $$\scriptsize y = x^4 – 5, \therefore \normalsize \frac{dy}{dx} \scriptsize = 4x^3 \; and \; \int 4x^3 dx = x^4 – 5$$

In the three examples we happen to show the expressions from which the derivative $$\scriptsize 4x^3$$was derived. But any constant term in the original expression becomes zero in the derivative and all trace of it is lost.

So if we do not know the history of the derivative $$\scriptsize 4x^3$$we have no evidence of the value of the constant term, be it 0,+2, -5, or any other value.

We, therefore, acknowledge the presence of such a constant term of some value by adding a symbol c to the result of the integration: i.e. $$\scriptsize \int 4x^3 dx = x^4 + c$$

c is called the constant of integration and must always be included. Such an integral is called an indefinite integral since normally we do not know the value of c.

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