Subject: Calculus

Arbitrary Constant Of Integration

Recall that in an indefinite integral, when the derivative of a function F is P, then the integral of P is given by

\int P(x)dx = F(x)+C

Now, the term C in the expression above is what we call the arbitrary constant of integration or simply constant of integration. The constant of integration is an optional element in indefinite integrals because of the non-uniqueness of these integrals as an inverse of derivative. The notion "optional element" does not mean that C is not an important part of indefinite integrals but is rather seen as a reason to maintain mathematical soundness among indefinite integrals.

Illustration

The reason why the arbitrary constant of integration is maintained in the evaluation of indefinite integrals is that the derivative of any constant is zero and remember that anti-differentiation is an accepted inverse of derivatives. So after evaluation of indefinite integrals and we want to go back the original function, we simply differentiate the right-hand side of the equation above and there we're back at the original function P without any sort of hindrance from the arbitrary constant C due to the fact that its derivative is zero and has nothing to do to alter the value of our integrand.
Consider the functions below and see how important C is to the indefinite integrals.

1.

f(x)=x^2

2.

g(x) = x^2 +1

3.

h(x)=x^2 +100

4.

r(x)=x^2 +999

Consider the functions f, g, h and r above. They are different, aren't they? But what happens if we differentiate these functions?

1.

f'(x)=D_{x}x^2 =2x

2.

f'(x)=D_{x}[x^2+1] =2x+0=2x

3.

f'(x)=D_{x}[x^2+100] =2x+0=2x

4.

f'(x)=D_{x}[x^2+999] =2x+0=2x

Now, the derivative of four different functions are exactly the same, isn't it? And if we want to go back to the original function, we just do the reverse process which is integration. Doing integration to the derivative above which is 2x gives x^2. Now, these should give us back to the original functions but obviously 2x is not the same to the functions g, h and r except f. The discrepancy is in the constants of the three functions g, h and r. So how do we compensate this discrepancy? Well, that's where the arbitrary constant of integration comes into play. By putting C into the integral of the derivatives of the functions, our integral now makes sense. C can be of any constant value. If we put C=1, then the function g is satisfied, if it is zero, then f is satisfied and so on.
In general, C is just an indication that any number added to F is an integral of the function P.

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