Antiderivatives - Integration, Business Mathematics & Statistics B Com Notes | EduRev

Business Mathematics and Statistics

Created by: Arshit Thakur

B Com : Antiderivatives - Integration, Business Mathematics & Statistics B Com Notes | EduRev

The document Antiderivatives - Integration, Business Mathematics & Statistics B Com Notes | EduRev is a part of the B Com Course Business Mathematics and Statistics.
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Antiderivatives

An antiderivative of a function f is a function whose derivative is f . In other words, F is an antiderivative of f if F' = f . To find an antiderivative for a function f , we can often reverse the process of differentiation.

For example, if f = x4, then an antiderivative of f is  Antiderivatives - Integration, Business Mathematics & Statistics B Com Notes | EduRev which can be found by reversing the power rule. Notice that not only is Antiderivatives - Integration, Business Mathematics & Statistics B Com Notes | EduRev an antiderivative of f , but so are   Antiderivatives - Integration, Business Mathematics & Statistics B Com Notes | EduRev etc. In fact, adding or subtracting any constant would be acceptable.


 

This should make sense algebraically, since the process of taking the derivative (i.e. going from F to f ) eliminates the constant term of F.

Because a single continuous function has infinitely many antiderivatives, we do not refer to "the antiderivative", but rather, a "family" of antiderivatives, each of which differs by a constant. So, if Fis an antiderivative of f , then G = F + c is also an antiderivative of f , and F and G are in the same family of antiderivatives.

 

Indefinite Integral

The notation used to refer to antiderivatives is the indefinite integral. Antiderivatives - Integration, Business Mathematics & Statistics B Com Notes | EduRev  f (x)dx means the antiderivative of f with respect to x . If F is an antiderivative of f , we can write Antiderivatives - Integration, Business Mathematics & Statistics B Com Notes | EduRev f (x)dx = F + c . In this context, c is called the constant of integration.

 

  1. x n dx =  Antiderivatives - Integration, Business Mathematics & Statistics B Com Notes | EduRev  x n+1 + c as long as n does not equal -1. This is essentially the power rule for derivatives in reverse
  2. Antiderivatives - Integration, Business Mathematics & Statistics B Com Notes | EduRev cf (x)dx = c Antiderivatives - Integration, Business Mathematics & Statistics B Com Notes | EduRev f (x)dx . That is, a scalar can be pulled out of the integral.
  3. Antiderivatives - Integration, Business Mathematics & Statistics B Com Notes | EduRev(f (x) + g(x))dx = Antiderivatives - Integration, Business Mathematics & Statistics B Com Notes | EduRev f (x)dx + Antiderivatives - Integration, Business Mathematics & Statistics B Com Notes | EduRev g(x)dx . The antiderivative of a sum is the sum of the antiderivatives.
  4. Antiderivatives - Integration, Business Mathematics & Statistics B Com Notes | EduRevsin(x)dx = - cos(x) + c 
    Antiderivatives - Integration, Business Mathematics & Statistics B Com Notes | EduRevcos(x)dx = sin(x) + c 
    Antiderivatives - Integration, Business Mathematics & Statistics B Com Notes | EduRevsec2(x)dx = tan(x) + c 
    These are the opposite of the trigonometric derivatives.

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