Integration (1 Viewer)

[Saintly Devil]

Which one is the answer to the indefinite integral of d(x^3)/dx (i.e. the integration of the differentiation of x cubed):

1) x^3 +C
2) x^3


I thought it was 2), since you already know the initial 'y' value, and so by integrating it's derivative, there is really no unknown.
But my teacher said it was x^3 + C. Can anyone explain why this is/isn't the case?

Thanks

 

[spice girl]

When you integrate something indefinitely, you get a whole set of answers that vary by the constant C. No exceptions.

An integral is not the exact opposite of a derivative.

 

[wogboy]

If you differentiate first, then integrate, you will need to include the constant of integration. If on the other hand, you integrate first, and then differentiate, you will not need to include the constant of integration. Why is that?

Take that example you gave:

d(x^3)/dx = 3x^2

Now if we integrate 3x^2 with respect to x, then we get:

x^3 + C

If however, we decide to integrate X^3 with respect to x first, then we get:

(x^4)/4 + C

If we differentiate this, then constant can be eliminated because the integration of any constant is zero.

so d([x^4]/4 + C)/dx = x^3

(with no constant)

When you differentiate a certain function of x, you are only worried about it's gradient (steepness) at a certain value of x, not it's absolute height, so as you differentiate, you actually lose information about that particular function of x, that you cannot get back simply by integrating it. Hence the purpose of putting C, which represents the uncertainty of the height of the original function of x.

 

[Saintly Devil]

Originally posted by spice girl
When you integrate something indefinitely, you get a whole set of answers that vary by the constant C. No exceptions.

An integral is not the exact opposite of a derivative.

Hmmmm.........thanks spice girl.

I think know now where i mis-understand the concept.