Integral Calculus

Integral Calculus

The Indefinite Integral

Copyright © 2005 by Ron Wallace, all rights reserved.

Antiderivative F(x) is an antiderivative of f(x) if F’(x) = f(x). Example: d  x 2  5x  3  2 x  5 dx

d  x 2  5x  7  2 x  5 dx

Therefore, x 2  5x  3 is an antiderivative of 2 x  5 Therefore, x 2  5x  7 is also an antiderivative of 2 x  5

Antiderivative

d Since c  0 for any constant c ... dx

d  x 2  5x  c   2 x  5 dx

Five antiderivatives of f(x)=2x-5 w/ c = 0, ±2, ±4

x  5x  c is an antiderivative of 2 x  5 2

Antiderivative Therefore:

If F’(x) = f(x), and c is any constant, then F(x) + c is an antiderivative of f(x).

Antiderivative Assume that F’(x) = f(x) and G’(x) = f(x). Then d/dx[F(x) - G(x)] = f(x) - f(x) = 0

Therfore F(x) - G(x) = c

So, antiderivatives of a function differ by a constant.

The Indefinite Integral The process of finding an antiderivative is called integration. Notation:

d  F ( x)  c  f ( x) dx

“The derivative of F(x)+c is f(x).”



“The indefinite integral of f(x) is F(x)+c.”

f ( x ) dx  F ( x )  c

Note that these two statements are different notations for the same fact (just opposite processes).

Integration Formulas Just reverse the differentiation formulas … x x e dx  e c 

x n 1  x dx  n  1  c, if n  1

x b x b  dx  ln b  c

n

  f ( x)  g ( x) dx   f ( x)dx   g ( x)dx

2 dx x2 dx 5x3 dx x3 (4x + 1)2 dx 7-2x3 dx 7-2x3 dx _____ x2

12 dx _____ (2x-3)2