why constant "c" used in integration?

Answer 1

An example: the derivative of x^2 (x squared) is 2x

the derivative of x^2 + 5 is also 2x because the derivative of 5 is 0

the c is put back in during integration to account for the possibility of the 5

when you are integrating over a range (like from 0 to 2) you can solve for the c and put in the actual number, but when you are just integrating in general you have to use the "c"

Answer 2

Differentiation of a constant C = 0

Integration is a reverse operation of differentiation.
So when you integrate you have to add a constant and solve to find the value. That constant could be 0 too.

Answer 3

Integration is the anti-derivative.

When you are given a function to find the integral, what you are doing is looking for a function that has the slope of the given equation.

If you are given y =2x and need to find the integral, what you are looking for is a function that has the slope = y=2x
at x=0, the slope is 0
at x=1, the slope is 2
at x=2, the slope is 4

The derivative of y = 2x is x^2 +C
By adding the constant C, the slope(derivative of x^2 +C) stays the same. The derivative of y =C is 0

y= x^2 +C
dy/dx =2x + 0 = 2x

Answer 4

The constant c does not matter when you are evaluating a definite integral. In fact, it does not even matter when you trying to find an indefinite integral. Example:

Suppose you have y = 3x^2,
then dy/dx = 6x.

Now if you integrate the derivative (i.e. 6x), you get 3x^2 + C. But wait, we started off without a constant? So as you can see, it does not matter. Academic asshol.s are anal retentive.

Answer 5

Suppose you want to find the anti-derivative of sin(x)cos(x).

If you let u=sin(x), du=cos(x)dx, and you will find the anti-derivative to be
(1/2) sin^2 (x)

If you let u=cos(x), then du=-sin(x)dx and you will find the anti-derivative to be
-(1/2) cos^2 (x)

If you rewrite the integrand as (1/2)sin(2x), you will find the anti-derivative to be
-(1/4)cos(2x).

ALL of these are anti-derivatives of the original function!

The point is that every function has infinitely many anti-derivatives. But, all of those anti-derivatives differ by constants from each other. That is the origin of the +C.

For example,
(1/2)sin^2(x)=-(1/2)cos^2(x) +1/2
=-(1/4)cos(2x) +1/4.

When doing definite integrals, the subtraction will cancel out any constant terms, so you only need to find one anti-derivative and stick to it. You will always get the same answer. But for indefinite integrals (where you want all the anti-derivatives), you need to show that the various answers can differ by that constant.

Answer 6

If y=f(x),
Then dy/dx is the slope of the tangent to the curve of f(x) at any point on the curve.
Sine if f(x)= g(x) + c ( c is any arbitrary constant )
Then f'(x) = g'(x)
So if we have the derivative of a function, there are many reverse derivative for that function different only in the constant which we denoted by c

Answer 7

When you do the derivative of any equation, any whole numbers or nonvariable numbers will become zero.
When you attempt to antiderive, you are finding the original equation from the equation of a derivative. Because a zero could be in the derivative equation from a constant in the original equation, a "c" is placed there to represent that nonvariable number in the antiderivative.

Hope it helps :)

Answer 8

when you mathematically integrate a function, you are integrating from point a to point b. the value of the integration only computes the area between the points, and thus is only a relative value. To know the exact value, you need to know where you are starting from. There is a good discussion at the link listed below.

http://en.wikipedia.org/wiki/Arbitrary_constant_of_integration#Where_does_the_constant_come_from.3F

Answer 9

Because during diffrentiation the integers are represented as zero. The integration is the reverse process of diffrentiation. no body exactly knows what was omitted during diffrentiation. hence constant is taken during integration and represented by first letter of constant i.e. "C"

Answer 10

because any constant could be there, since the derivative of a constant is zero. there is no way of knowing whether a constant should be zero or a number.