Introduction to Evaluation of Integrals

 

 

 

1.

 

Basic integrals you need to know are for the following functions.

 

a.         polynomials

b.        sine and cosine functions

c.         exponential functions

 

 

 

 

 

 

 

 

2.

 

Idefinite integrals (no limits of integration are specified) that you MUST know:

 

   Terminology:  the “integrand” is the function being integrated

 

                             Integral                            Integrand

a.         ∫ xn dx =    (xn+1)/(n+1)  + c                       xn           with restriction  n ≠ -1       

 

b.        ∫ sin x dx =  - cox x  + c                           sin x

 

    c.    ∫ cos x dx =    sin x  + c                            cos x

 

   d.    ∫ eax dx     =    eax  / a   + c                          eax 

 

where  c  is the constant of integration

 

 

3.

 

Note:  Differentiation of the result of the integration should return the integrand.

           So you can always check to see if your result for integration is correct.

 

 

 

 

 

4.

 

 

Example:     I   =  ∫(x + 1)2 dx      =   ∫ [ x2  +  2 x  +  1 ] dx

 

                     I  =  x3 / 3  +  x2  +  x   +  c

 

  Check      dI/dx  =  x2  +  2 x  +  1  =  (x + 1)2    which yields the integrand; 

  So this integration was correct.

 

  Here   I  is called an “indefinite” integral and    c   is the “constant of integration”

 

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 Copyright © 2009 Richard C. Coddington

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