Integrals Involving Quadratic Polynomials

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.

 

In a Nut Shell:  A quadratic polynomial has the form    ax2  + bx  + c.

The strategy to evaluate simple integrals involving quadratic polynomials is to

complete the square followed by an appropriate trigonometric substitution

as illustrated by the example shown below.

 

 

Example:    I   =   ∫ dx / (x2  + 2x  + 5)

 

Complete the square with     (x + 1)2  +  4  =  (x + 1)2  +  22

 

  I  =    ∫ dx / [ (x + 1)2  +  22 ]

 

Now let    u  =  x + 1,  du  =  dx

 

  I  =       ∫ du / ( u2 + 22 )   which suggests the following trig substitution

 

        Let    u = 2 tan θ,  du  =  2 sec2 θ  d θ,  u2 + 22  =   22 sec2 θ

 

 

  I   =    ∫ 2 sec2 θ  d θ / 22 sec2 θ  =  ∫ (1/2) d θ  =  (1/2) θ  +  C

 

  Now   θ  =  tan-1 (u/2)  =  tan-1 [( x+1)/2] 

 

So the integral becomes:

 

 

             I  =  (1/2) tan-1 [( x+1)/2]  +  C

 

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

was correct.  Here’s the check:

 

dI/dx  =  (1/2) [ 1/ {(1+ ( x+1)2/22}(1/2) ] =1/ [x2  + 2x  + 1 + 4]

 

dI/dx  = 1/( x2  + 2x  + 5)    check!

 

Click here for a more complicated example.

 

 Copyright © 2011 Richard C. Coddington

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