Friday, 25 April 2014

calculus - Purpose Of Adding A Constant After Integrating A Function



I would like to know the whole purpose of adding a constant termed constant of integration everytime we integrate an indefinite integral f(x)dx. I am aware that this constant "goes away" when evaluating definite integral baf(x)dx. What has that constant have to do with anything? Why is it termed as the constant of integration? Where does it come from?



The motivation for asking this question actually comes from solving a differential equation xdydx=5x3+4 By separation of dy and dx and integrating both sides, dy=(5x2+4x)dx yields y=5x33+4ln(x)+C.



I've understood that dy represents adding infinitesimal quantity of dy's yielding y but I'am doubtful about the arbitrary constant C.


Answer



Sometimes you need to know all antiderivatives of a function, rather than just one antiderivative. For example, suppose you're solving the differential equation
y=4y
and there's an initial condition y(0)=5. You get
dydx=4y
dyy=4dx
dyy=4dx
logey=4x+C.
Here you've added a constant. Then
y=e4x+C=e4xeC
5=y(0)=e40eC=eC
So
y=5e4x.



(And sometimes you need only one antiderivative, not all of them. For example, in integration by parts, you may have dv=cosxdx, and conclude that v=sinx.)


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