Basic Methods of Integration

 

A.

Memorize the Standard Form Integrals - - This is mandatory.  Standard integrals

include integrals involving functions such as polynomials, exponentals, trig functions,

combinations of these and others.

 

B.

Memorize identities.  i.e. For Trig identities click here.

 

 

 

C.

 

                ****** Key  Strategy to Evaluate Integrals ******

Transform more complicated integrals (not in standard form)

into a form identical to one of the standard form integrals.

 

There are several procedures to accomplish this transformation.

 

 

 

1.

Use simple substitutions.

 

Example:     ∫(x + 1)2 dx             let  u = x + 1,       du  =  dx

 

    Then the integral becomes    ∫u2 du    which is of

     the standard form                             ∫ xn dx = (xn+1)/(n+1)  + c

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2.

Multiple Use of Simple Substitutions

 

Example:     I   =   ∫ √(4 – x2) dx       let   x = 2 sin θ,       dx  = 2 cos θ  d θ

 

   4 – x2  =  4 – 22  sin2 θ  =  4 cos2 θ,    so  √4 cos2 θ  =  2 cos θ 

 

Then the integral becomes    4 ∫ cos2 θ d θ

 

Next use the substitution    cos2 θ = (1 + cos2θ )/2

 

So the integral becomes    4/2  ∫(1 + cos2θ ) d θ 

 

Or  I  =  2 ∫d θ   and   2 ∫cos2θ d θ 

 

which results in two standard integrals:

 

     ∫ dθ  =  θ and    ∫cos 2θ  dx  = [sin 2θ]/2   When finished you still need to convert

 

    your result back into terms of the original variable, x.

 

  i.e.  θ  =  sin-1(x/2)    and   sin 2θ  =  2 sin θ cos θ      and

 

     sin θ   =  x/2,            cos θ  =  √(4 – x2)

 

      I  = 2 [sin-1(x/2)]    +      [x/2]  [√ (22 – x2 )]    +  C

 


Copyright © 2008 Richard C. Coddington

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